a correction method of estimating the pointing...
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Research ArticleA Correction Method of Estimating the PointingError for Reflector Antenna
Jie Zhang 12 Jin Huang 1 Wei Liang1 Youran Zhang1 Qian Xu2
Letian Yi2 and Congsi Wang 1
1Key Laboratory of Electronic Equipment Structure Design Ministry of Education Xidian University Xirsquoan 710071 China2Xinjiang Astronomical Observatory Chinese Academy Xinjiang 830011 China
Correspondence should be addressed to Jin Huang jhuangmailxidianeducn and Congsi Wang congsiwang163com
Received 15 December 2017 Accepted 22 April 2018 Published 30 May 2018
Academic Editor Francesco Franco
Copyright copy 2018 Jie Zhang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The pointing error caused by the structural deformation of large reflector antennas has become the most challenging problem ofantenna servo control Especially with the increase of the antenna diameter and the working frequency environmental loads notonly make the structure deformationmore obvious but alsomake the pointing accuracy influenced by deformation more sensitiveIn order to solve this problem accurately estimating the pointing error caused by the structural deformation is the key Based on thedynamic model of antenna structure and the analyzing model of pointing error using the displacement information of samplingpoints on reflector this paper proposes a correction method to achieve the purpose of accurately estimating the pointing errorcaused by the structural deformation Using a 73-m Ka band antenna the results show that the antenna maximum pointing errorin theoretical model calculation is 00041∘ at 10ms wind speed condition however the corrected pointing error would be about00054∘ with considering themodeling error After compensating the controller the pointing error could be reduced to only 00008∘and the performance of antenna pointing was improved
1 Introduction
Thepointing accuracy is one of the most important technicalperformance indicators of radio telescope which is the maininstrument to receive and transmit electromagnetic wavesQitai 110m radio telescope (QTT) which will be built in Xin-jiang China has pointing accuracy required to reach 00004∘ However due to the environmental load the pointing errorwould be different as follows It is about 0011∘ caused bythe deformation from the gravity distribution under differentconditions It is about 0004∘ caused by the deformation fromtemperature gradient And it is up to 0024∘ caused by thedeformation from random wind disturbance [1] These arefar beyond its pointing accuracy requirements Thereforethe accurate estimation of the pointing errors caused by theenvironmental load is the basis for guaranteeing pointingaccuracy of the large reflector antenna
The study of pointing errors caused by structural defor-mation under different conditions such as temperature and
gravity distribution is relatively mature because the changeof the loads is slow The laser measurement techniques wereused to obtain the root mean square error (RMSE) of theantenna reflector deformation at different pitch angles andestimate the effect of gravity on the deformation of thestructure [2] By using photogrammetric technology theantenna reflecting surface deformation of 22m antenna atdifferent pitch angles was obtained and its impact on theelectrical performance of antenna pointing was analyzed [3]For the deformation of the reflecting surface a structuraloptimization of large reflector antennas was proposed whichcould reduce the effect of the deformation caused by gravityon the pointing accuracy [4] Based on the load analysis ofthe finite element model the reflector deformation infor-mation was obtained And making use of the adjustment ofsubreflector position the effective compensation was realizedto reduce the antenna pointing error [5] The subreflectorarray adjustment was used to achieve the match of thesubreflector shape and to compensate for the loss of electrical
HindawiShock and VibrationVolume 2018 Article ID 3262869 12 pageshttpsdoiorg10115520183262869
2 Shock and Vibration
performance whichwas caused by themain surface deforma-tion from the slow changes such as gravity and temperature[6] With considering the effect of temperature on thestructural deformation the optimal structure of reflectorback frame was obtained by use of topology optimizationwhich could realize the effective inhibition to structuraldeformation caused by temperature gradient [7 8] An elec-tromechanical coupling analysis method of electromagneticperformance was proposed for the 40m antenna structuraldeformation in the solar radiation [9] The result shows thatthe electromagnetic performance parameters mainly dependon the reflecting surface deformation distribution not theroot mean square error of the reflecting surface deformation
Because of slow change and high repeatability of tem-perature and gravity effects it is possible to set up the errortable offline whether using finite element software analysis oractual sensors measurement Thereby it could be overcomeby look-up table method in the process of antenna servocontrol However as a random transient load which is moredifficult to analyze and inhibit the wind disturbance has alsoreceived widespread attention
The servo error data produced by the wind disturbanceof specific antenna location was collected and the equivalenttorque coefficient of wind disturbance at equivalent distur-bance torque was derived which played a guiding role in thewind disturbance simulation in other antenna design [10]According to the characteristics of wind disturbance on theantenna the influence of wind disturbance on the antennapointing was introduced into the simulation analysis whichcould be equivalent to three kinds of disturbance formswhichare the wind disturbance acting on the reflector surface ofthe antenna the torque disturbance acting on the motorshaft of the antenna and the speed disturbance acting onthe velocity loop input [11] All of them have achieved goodsimulation results The finite element software was used toanalyze the structural deformation of the 40m antenna atdifferent wind speeds and the multi-field coupled modelwas proposed to analyze the pointing error caused by thestructural deformation [12] An antenna dynamic model wasintroduced to analyze the influence of the deformation ofdifferent structures on the pointing error The result revealedthat the pointing error caused by the flexible deformationof the antenna cannot be ignored [13] Using the beamwaveguide system a fast active compensation structure wasdesigned and through the optimized selection of the ampli-fication factor it is possible to quickly compensate for thepointing error caused by wind disturbance [14]
In recent years although the pointing error of thestructural deformation caused by wind disturbance hasgradually received attention The analysis method could beeither obtaining structural deformation by the finite elementsoftware simulation to estimate the pointing error howeverit cannot be directly applied to the controller implemen-tation or establishing pointing error analysis model basedon the modal superposition method and the approximateoptical method but this method depends on the accuracyof the model which is hard to guarantee for the loadmodel error from the complicate wind pressure distributionon the antenna reflecting surface caused by the turbulent
Modal analysis of antenna structure
The dynamic modelof antenna structure
The analysis of pointing error
e nodal displacements
The correctionmethod of pointing
error
The actual displacements of sampling nodes
Obtaining optimalcorrection weight
factor
e actual pointing error
Figure 1 The flow chart of the correction method
Reflector system Central hub
Mount bases
Azimuth rotation
Z
XY
Elevation rotation
Figure 2 The structure of dual reflector antenna
characteristics of the wind field and the dynamic model errorIt often brings a burden to the design of the controller
This paper presents a correction method of estimating thepointing error caused by flexible deformation of antennasThe flow chart of this algorithm is shown as Figure 1 basedon the establishment of dynamic model of antenna structureand the analysis of pointing error it constructs pointing errorcorrection method according to the actual displacementsof sampling nodes of reflection surface By modifying theoptimal correction weight factor it eventually achieves thepurpose of accurate estimation of the pointing error
2 The Dynamic Model of Antenna Structure
To analyze the pointing error caused by the environmentalload the first step is to obtain the deformation of theantenna structure under the environmental load As shown inFigure 2 taking a large dual reflector antenna as an examplethe structure consists of the mount base the central huband the reflector system Any deformation of the parts of theantenna structure will lead to the deflection of the antennapointing The deformation of the mount base and the centralhub can eventually be reflected in the deflection of thereflector system As long as we obtain the final position of
Shock and Vibration 3
each node of the reflector system the antenna error causedby the deformation of the structure can be syntheticallyestimated
The dynamic model of antenna structure under general-ized coordinates can be expressed as follows [15]
Mq +Dq + Kq = B0u (1)
y = Coq (2)
where the diagonal matrices M K and D are the massmatrix stiffness matrix and damping matrix respectivelyThe environment load inputs of the flexible model is 119906 Theoutput vector y is obtained by the nodal displacement vectorq B0 is the input matrix and C0 is the output matrix
To express the structure in modal coordinates the modaldisplacement is introduced which satisfies the followingequations
q = Φqm (3)
Φ is the modal shape matrix consisting of the 119895119905ℎ nodedisplacement of the 119894119905ℎmode 120601119894119895
Φ =[[[[[[[[[[[[
12060111 12060121 sdot sdot sdot 120601n112060112 12060122 sdot sdot sdot 120601n2sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1206011k 1206012k sdot sdot sdot 120601nksdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1206011nd1206012nd
sdot sdot sdot 120601nnd
]]]]]]]]]]]]
(4)
Introducing (3) into (1) and (2) and multiplying by ΦTthe dynamic equation based on modal coordinate could beobtained
Mmqm + Dmqm + Kmqm = ΦTB0u (5)
y = CoΦqm (6)
The mass matrix stiffness matrix and damping matrix weretransformed into modal mass matrix Mm modal stiffnessmatrix Km and modal damping matrix Dm respectively
Mm = ΦTMΦ (7)
Km = ΦTKΦ (8)
Dm = ΦTDΦ (9)
Meanwhile the input matrix and output matrix are trans-formed into modal input matrix Bm and modal outputmatrix Cm
ΦTB0 = Bm = [[[[[[
11988711989811198871198982sdot sdot sdot119887119898119894]]]]]]
(10)
CoΦ = Cm = [1198881198981 1198881198982 sdot sdot sdot 119888119898119894] (11)
Design main reflector
Deformed main reflector
OSub-reflector
B N
U
Wdb dn
B㰀 Feed
N㰀 휃s
Figure 3 The factors influencing the performances of antennapointing
Themodal mass matrix andmodal stiffness matrix satisfythe following equation where Ω is the natural frequencymatrix and 120596119894 is the frequency of 119894119905ℎ order
Ω2 = Mmminus1Km (12)
Ω = [[[[[[
1205961 0 sdot sdot sdot 00 1205962 sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot 120596n]]]]]] (13)
The modal damping ratio matrix Z could be expressedas (14) and the transfer function of the 119894119905ℎmode is obtainedfrom (15)
Z = [[[[[[
1205851 0 sdot sdot sdot 00 1205852 sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot 120585n]]]]]]= 05Mm
minus1DmΩminus1 (14)
119892119897119894 (120596) = 1198881198981198941198871198981198941205961198942 minus 1205962 + 2119895120585119894120596119894120596 (15)
3 The Analysis of Pointing Error Caused byStructure Deformation
To dual reflector antenna the factors influencing the per-formances of antenna pointing include the deformation ofthe main reflector the lateral displacement (perpendicular tothe focal axis) of the feed the lateral displacement and therotation of the subreflector as shown in Figure 3
4 Shock and Vibration
When analyzing the influence of the deformed mainreflector on pointing error the best fitting paraboloid shouldbe obtained based on the displacement of each node ofthe reflector There are six key parameters to confirmthe best fitting paraboloid They are the displacement ofthe vertex of the fitted parabolic reflector 119889119906 119889V 119889119908 therotation angles of the focal axis of the fitted parabolic
reflector 120601119906 120601V and the variation in the focal length119889119891The relationship among these parameters and the dis-
placement of each node could be expressed as follows [16]
119860119899120573 = 119867119899 (16)
where
119860119899 =
[[[[[[[[[[[[[[[[
119899sum119904=1
11990611990422119891119899sum119904=1
119906119904V1199042119891 minus 119899sum119904=1
119906119904 minus 119899sum119904=1
119906119904V119904 119899sum119904=1
1199061199042 119899sum119904=1
119906119904119908119904119891119899sum119904=1
119906119904V1199042119891119899sum119904=1
V1199042
2119891 minus 119899sum119904=1
V119904 minus 119899sum119904=1
V1199042119899sum119904=1
119906119904V119904 119899sum119904=1
V119904119908119904119891119899sum119904=1
1199061199041199081199042119891119899sum119904=1
V1199041199081199042119891 minus 119899sum119904=1
119908119904 minus 119899sum119904=1
V119904119908119904 119899sum119904=1
119906119904119908119904 119899sum119904=1
1199081199042119891119899sum119904=1
1199061199042119891119899sum119904=1
V1199042119891 minus119899 minus 119899sum119904=1
V119904119899sum119904=1
119906119904 119899sum119904=1
119908119904119891
]]]]]]]]]]]]]]]]
(17)
119867119899 = [ 119899sum119904=1(119908119904 minus 1199081199041015840) 119906119904 119899sum
119904=1(119908119904 minus 1199081199041015840) V119904 119899sum
119904=1(119908119904 minus 1199081199041015840)119908119904 119899sum
119904=1(119908119904 minus 1199081199041015840)]119879 (18)
120573 = [119889119906 119889V 119889119908 120601119906 120601V 119889119891]119879 (19)
where 119906119904 V119904 119908119904 is the design coordinate of 119904119905ℎ node1199081199041015840 isthe coordinate after deformation along the direction of focalaxis and 119891 is the length of the focal axis
As shown in Figure 4 after obtaining the best fittingparaboloid the pointing error in azimuth direction causedby the lateral displacement of the vertex 119889119906 the rotationangle of the focal axis 120601V the lateral displacement of the feed119889119887 the lateral displacement of the subreflector 119889119899 and therotation angle of the subreflector 120579119904 could be obtained usingthe following equations
The distance between B1015840 and the focal axis of the subre-flector is 1198891199041 which leads to a lateral displacement 1198891199042
1198891199041 = cos 120579119904 1003816100381610038161003816119889119887 + 119889119899 + tan 12057911990411987111003816100381610038161003816 (20)
1198891199042 = 119889119904111987131198711 (21)
where 1198711 is the distance between points B and N and 1198713is the distance between points N1015840 and F119870119908 is the beam deviation factor which is related to theratio of the paraboloid focal distance to its open diameter andaperture-field distribution By omitting the vertex displace-ment of the paraboloid along the W axis the actual pointingerror caused by structural deformation can be expressed as120579119899
119889119900 = 1003816100381610038161003816tan 120601V (1198711 + 1198713 cos 120579119904 minus 1198891199042 sin 120579119904) minus (119889119899 + 1198713 sin 120579119904 + 1198891199042 cos 120579119904) + tan 120601V1198712 minus 1198891199061003816100381610038161003816radic1 + (tan 120601V)2 (22)
120579119889 = 120601V minus 119889119900119870119908119891 (23)
4 Correction Method of Estimatingthe Pointing Error
During the estimation of the pointing error 120579119899 it ishard to accurately obtain the input of the environment
loads because of the complexity of wind field Mean-while taking the modeling error of dynamic model intoaccount it usually leads to some unavoidable estimationerror of the pointing error As a result 120579119899 needs to becorrected
Shock and Vibration 5
Design paraboloid
Fitting paraboloid
U
do
휃d
B
db ds1
B㰀
휃u
휃s
dn
N
휙
F㰀
F
du
ds2
W
Figure 4 The pointing error caused by reflector deformation
From (6) the node displacement could be obtained bythe superposition of displacement of each mode so thedisplacement of 119897119905ℎ node could be expressed as follows
119904119897 = 119899sum119894=1
119862119900Φ119902119894 = 119899sum119894=1
119910119897119894 (24)
where 119910119897119894 is the displacement of the 119894119905ℎ mode So thepointing error could also be expressed as follows
120579119889 = 119899sum119894=1
120579119894 (25)
where 120579119894 is the pointing error of 119894119905ℎmode Assuming theactual pointing error 1205791198891015840 is obtained by
1205791198891015840 = 119899sum119894=1
(1 + 120572119894) 120579119894 (26)
where 120572119894 is correction weight factor of each mode Theactual displacement 1199041198971015840 of 119897119905ℎ node could be obtained
1199041198971015840 = 119899sum119894=1
(1 + 120572119894120574119894 )119910119897119894 (27)
where 120574119894 is the gain from node displacement to pointingerror of 119894119905ℎmode it could be defined as follows
120574119894 = 120579119894119910119897119894 = 120579119894119906 times 119906119910119897119894 = 100381710038171003817100381711989211991111989410038171003817100381710038172 times 1100381710038171003817100381711989211989711989410038171003817100381710038172 (28)
119892119911119894 is the transfer function from environment loadsto pointing error and 119892119897119894 is the transfer function fromenvironment loads to node displacement as shown in(15) sdot 2 denotes the H2 norm of the transfer func-tion
To simplify the pointing error expression omitting theeffect of the high-order term as (29) 119892119911119894 could be obtainedfrom (30)
1198891199042120579119904 asymp 0 (29)
119892119911119894 = (Aminus151119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)minus 1003816100381610038161003816100381610038161003816100381610038161003816(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)(1198711 + 1198713) minus (119892119894119861+ 1198713119892119894120579 + 1198713 1003816100381610038161003816119892119894119873 + 119892119894119861 + 119871111989211989412057910038161003816100381610038161198711 )+ 1198712(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894) minus (Aminus111119899sum119897=1
119892119897119894119906119904+ Aminus112
119899sum119897=1
119892119897119894V119904 + Aminus113119899sum119897=1
119892119897119894119908119904 + Aminus114119899sum119897=1
119892119897119894)1003816100381610038161003816100381610038161003816100381610038161003816sdot 119870119908119891
(30)
where 119892119894119861 is the transfer function from the environ-ment loads to the displacement of the feed of the 119894119905ℎmode 119892119894119873 is the transfer function from the environmentloads to the displacement of the subreflector of the 119894119905ℎmode and 119892119894120579 is the transfer function from the environ-ment loads to the rotation angel of the subreflector of 119894119905ℎmode
[Aminus151 Aminus152 Aminus153 Aminus154]= [0 0 0 0 1 0]An
minus1(31)
[Aminus111 Aminus112 Aminus113 Aminus114]= [1 0 0 0 0 0]An
minus1(32)
6 Shock and Vibration
The H2 norm of each transfer function could be derivedrespectively
100381710038171003817100381711989211989711989410038171003817100381710038172 = 100381710038171003817100381711988811989811989410038171003817100381710038172 1003817100381710038171003817119887119898119894100381710038171003817100381722radic120585119894120596119894 (33)
100381710038171003817100381711989211991111989410038171003817100381710038172 asymp 100381610038161003816100381611988711989811989410038161003816100381610038162radic120585119894119908119894 (1 +(1198711 + 1198713)119870119908119891 + 119870119908119891 )
sdot radic(Aminus151)2 119899sum119904=1
(119888119898119902119894119904119906119904)2 + (Aminus152)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus153)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus154)2 119899sum119904=1
(119888119898119902119894119904)2 + 119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic1198881198981199021198941198732 + 119871321198881198981199021198941205792 + 1198701199081198713 1003816100381610038161003816119887119898119894100381610038161003816100381621198911198711radic120585119894119908119894radic1198881198981199021198941198612 + 1198881198981199021198941198732 + 119871121198881198981199021198941205792 +
119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic(Aminus111)2 119899sum
119904=1
(119888119898119902119894119904119906119904)2 + (Aminus112)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus113)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus114)2 119899sum119904=1
(119888119898119902119894119904)2
(34)
where 119888119898119902119894119861 is the 119894119905ℎmode of node B which is the centralpoint of the feed 119888119898119902119894119873 is the 119894119905ℎ mode of the modal outputmatrix of nodeNwhich is the central point of the subreflectorand 119888119898119902119894120579 is the 119894119905ℎmode of the modal output matrix which isfor deriving the rotation angle of the subreflector
When the node displacements 1199041 1199042 119904119897 of the lim-ited sampling points are obtained to obtain the optimalcorrection weight factor the sum 120575119904 given in the followingshould have its smallest value And the actual pointingerror 1205791198891015840 could be derived after the correction weight factordetermined
120575119904 = radicsum119897119894=1 (119904119894 minus 1199041198941015840)2
119897 (35)
5 Model Application
Taking the antenna with a diameter of 73 meters as anexample the antenna and its finite element model are shownin Figure 5 The finite element model is used to derive modalinformation (modal matrices) of the structure which lays afoundation for the dynamic model What is more some testsare used to prove that the finite element model of the antennawas compliantwith a real antenna and some analytical resultsindicate that the proposed model is effective Next it isexplained in detail
51 Experimental Testing In order to verify the accuracy ofthe finite element model a frequency test experiment and aload deformation experiment are made to the antenna
A load deformation experiment is applied by the 100Kgimpulse in lateral loading and vertical loading respectivelyAs shown in Figure 6 red arrows represent load actiondirection Deformation testing equipment is the laser tracker
(API) An example of field testing is shown in Figure 7Numeral 1 represents the API laser tester Numeral 2 repre-sents the target location Numeral 3 represents 100Kg loadThree sets of testing were done at elevation angles of 30∘ 50∘and 70∘ respectively Comparing the ANSYS finite elementsimulation results and API test results the static maximumdeformation and the pointing error in the deformationprocess are shown in Table 1 The relative error representsthe ratio of the absolute pointing error and the test results ofpointing error
Take the dynamic process into account taking elevationangle of 70∘ for example API test results and simulationresults are compared as shown in Figure 8
It can be seen from the comparison of the results thatstatic load deformation error is less than 20 and dynamicresponse characteristics are basically consistent
Frequency test experiments were performed in two direc-tions by loading impulse loads (to generate oscillations ofthe antenna structure in the two directions) The oscillationdirections corresponding to orientation and elevation wereadopted and accelerometer sensors were placed on the twopoints with the largest amplitudes Then by using the modalanalyzer the natural frequencies in the two directions wereacquired by analyzing the data collected by the accelerometersensors and comparing the results with those of the ANSYSmodel with the same oscillation modal (with the sameoscillation) Table 2 shows a comparison of the test results andsimulation results
From the results of frequency test experiment and loaddeformation experiment it is considered that the finiteelement analysis results basically meet the accuracy require-ment The finite element model can be used as a basis fordynamic modeling and reference for real-testing structuraldeformation
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
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2 Shock and Vibration
performance whichwas caused by themain surface deforma-tion from the slow changes such as gravity and temperature[6] With considering the effect of temperature on thestructural deformation the optimal structure of reflectorback frame was obtained by use of topology optimizationwhich could realize the effective inhibition to structuraldeformation caused by temperature gradient [7 8] An elec-tromechanical coupling analysis method of electromagneticperformance was proposed for the 40m antenna structuraldeformation in the solar radiation [9] The result shows thatthe electromagnetic performance parameters mainly dependon the reflecting surface deformation distribution not theroot mean square error of the reflecting surface deformation
Because of slow change and high repeatability of tem-perature and gravity effects it is possible to set up the errortable offline whether using finite element software analysis oractual sensors measurement Thereby it could be overcomeby look-up table method in the process of antenna servocontrol However as a random transient load which is moredifficult to analyze and inhibit the wind disturbance has alsoreceived widespread attention
The servo error data produced by the wind disturbanceof specific antenna location was collected and the equivalenttorque coefficient of wind disturbance at equivalent distur-bance torque was derived which played a guiding role in thewind disturbance simulation in other antenna design [10]According to the characteristics of wind disturbance on theantenna the influence of wind disturbance on the antennapointing was introduced into the simulation analysis whichcould be equivalent to three kinds of disturbance formswhichare the wind disturbance acting on the reflector surface ofthe antenna the torque disturbance acting on the motorshaft of the antenna and the speed disturbance acting onthe velocity loop input [11] All of them have achieved goodsimulation results The finite element software was used toanalyze the structural deformation of the 40m antenna atdifferent wind speeds and the multi-field coupled modelwas proposed to analyze the pointing error caused by thestructural deformation [12] An antenna dynamic model wasintroduced to analyze the influence of the deformation ofdifferent structures on the pointing error The result revealedthat the pointing error caused by the flexible deformationof the antenna cannot be ignored [13] Using the beamwaveguide system a fast active compensation structure wasdesigned and through the optimized selection of the ampli-fication factor it is possible to quickly compensate for thepointing error caused by wind disturbance [14]
In recent years although the pointing error of thestructural deformation caused by wind disturbance hasgradually received attention The analysis method could beeither obtaining structural deformation by the finite elementsoftware simulation to estimate the pointing error howeverit cannot be directly applied to the controller implemen-tation or establishing pointing error analysis model basedon the modal superposition method and the approximateoptical method but this method depends on the accuracyof the model which is hard to guarantee for the loadmodel error from the complicate wind pressure distributionon the antenna reflecting surface caused by the turbulent
Modal analysis of antenna structure
The dynamic modelof antenna structure
The analysis of pointing error
e nodal displacements
The correctionmethod of pointing
error
The actual displacements of sampling nodes
Obtaining optimalcorrection weight
factor
e actual pointing error
Figure 1 The flow chart of the correction method
Reflector system Central hub
Mount bases
Azimuth rotation
Z
XY
Elevation rotation
Figure 2 The structure of dual reflector antenna
characteristics of the wind field and the dynamic model errorIt often brings a burden to the design of the controller
This paper presents a correction method of estimating thepointing error caused by flexible deformation of antennasThe flow chart of this algorithm is shown as Figure 1 basedon the establishment of dynamic model of antenna structureand the analysis of pointing error it constructs pointing errorcorrection method according to the actual displacementsof sampling nodes of reflection surface By modifying theoptimal correction weight factor it eventually achieves thepurpose of accurate estimation of the pointing error
2 The Dynamic Model of Antenna Structure
To analyze the pointing error caused by the environmentalload the first step is to obtain the deformation of theantenna structure under the environmental load As shown inFigure 2 taking a large dual reflector antenna as an examplethe structure consists of the mount base the central huband the reflector system Any deformation of the parts of theantenna structure will lead to the deflection of the antennapointing The deformation of the mount base and the centralhub can eventually be reflected in the deflection of thereflector system As long as we obtain the final position of
Shock and Vibration 3
each node of the reflector system the antenna error causedby the deformation of the structure can be syntheticallyestimated
The dynamic model of antenna structure under general-ized coordinates can be expressed as follows [15]
Mq +Dq + Kq = B0u (1)
y = Coq (2)
where the diagonal matrices M K and D are the massmatrix stiffness matrix and damping matrix respectivelyThe environment load inputs of the flexible model is 119906 Theoutput vector y is obtained by the nodal displacement vectorq B0 is the input matrix and C0 is the output matrix
To express the structure in modal coordinates the modaldisplacement is introduced which satisfies the followingequations
q = Φqm (3)
Φ is the modal shape matrix consisting of the 119895119905ℎ nodedisplacement of the 119894119905ℎmode 120601119894119895
Φ =[[[[[[[[[[[[
12060111 12060121 sdot sdot sdot 120601n112060112 12060122 sdot sdot sdot 120601n2sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1206011k 1206012k sdot sdot sdot 120601nksdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1206011nd1206012nd
sdot sdot sdot 120601nnd
]]]]]]]]]]]]
(4)
Introducing (3) into (1) and (2) and multiplying by ΦTthe dynamic equation based on modal coordinate could beobtained
Mmqm + Dmqm + Kmqm = ΦTB0u (5)
y = CoΦqm (6)
The mass matrix stiffness matrix and damping matrix weretransformed into modal mass matrix Mm modal stiffnessmatrix Km and modal damping matrix Dm respectively
Mm = ΦTMΦ (7)
Km = ΦTKΦ (8)
Dm = ΦTDΦ (9)
Meanwhile the input matrix and output matrix are trans-formed into modal input matrix Bm and modal outputmatrix Cm
ΦTB0 = Bm = [[[[[[
11988711989811198871198982sdot sdot sdot119887119898119894]]]]]]
(10)
CoΦ = Cm = [1198881198981 1198881198982 sdot sdot sdot 119888119898119894] (11)
Design main reflector
Deformed main reflector
OSub-reflector
B N
U
Wdb dn
B㰀 Feed
N㰀 휃s
Figure 3 The factors influencing the performances of antennapointing
Themodal mass matrix andmodal stiffness matrix satisfythe following equation where Ω is the natural frequencymatrix and 120596119894 is the frequency of 119894119905ℎ order
Ω2 = Mmminus1Km (12)
Ω = [[[[[[
1205961 0 sdot sdot sdot 00 1205962 sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot 120596n]]]]]] (13)
The modal damping ratio matrix Z could be expressedas (14) and the transfer function of the 119894119905ℎmode is obtainedfrom (15)
Z = [[[[[[
1205851 0 sdot sdot sdot 00 1205852 sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot 120585n]]]]]]= 05Mm
minus1DmΩminus1 (14)
119892119897119894 (120596) = 1198881198981198941198871198981198941205961198942 minus 1205962 + 2119895120585119894120596119894120596 (15)
3 The Analysis of Pointing Error Caused byStructure Deformation
To dual reflector antenna the factors influencing the per-formances of antenna pointing include the deformation ofthe main reflector the lateral displacement (perpendicular tothe focal axis) of the feed the lateral displacement and therotation of the subreflector as shown in Figure 3
4 Shock and Vibration
When analyzing the influence of the deformed mainreflector on pointing error the best fitting paraboloid shouldbe obtained based on the displacement of each node ofthe reflector There are six key parameters to confirmthe best fitting paraboloid They are the displacement ofthe vertex of the fitted parabolic reflector 119889119906 119889V 119889119908 therotation angles of the focal axis of the fitted parabolic
reflector 120601119906 120601V and the variation in the focal length119889119891The relationship among these parameters and the dis-
placement of each node could be expressed as follows [16]
119860119899120573 = 119867119899 (16)
where
119860119899 =
[[[[[[[[[[[[[[[[
119899sum119904=1
11990611990422119891119899sum119904=1
119906119904V1199042119891 minus 119899sum119904=1
119906119904 minus 119899sum119904=1
119906119904V119904 119899sum119904=1
1199061199042 119899sum119904=1
119906119904119908119904119891119899sum119904=1
119906119904V1199042119891119899sum119904=1
V1199042
2119891 minus 119899sum119904=1
V119904 minus 119899sum119904=1
V1199042119899sum119904=1
119906119904V119904 119899sum119904=1
V119904119908119904119891119899sum119904=1
1199061199041199081199042119891119899sum119904=1
V1199041199081199042119891 minus 119899sum119904=1
119908119904 minus 119899sum119904=1
V119904119908119904 119899sum119904=1
119906119904119908119904 119899sum119904=1
1199081199042119891119899sum119904=1
1199061199042119891119899sum119904=1
V1199042119891 minus119899 minus 119899sum119904=1
V119904119899sum119904=1
119906119904 119899sum119904=1
119908119904119891
]]]]]]]]]]]]]]]]
(17)
119867119899 = [ 119899sum119904=1(119908119904 minus 1199081199041015840) 119906119904 119899sum
119904=1(119908119904 minus 1199081199041015840) V119904 119899sum
119904=1(119908119904 minus 1199081199041015840)119908119904 119899sum
119904=1(119908119904 minus 1199081199041015840)]119879 (18)
120573 = [119889119906 119889V 119889119908 120601119906 120601V 119889119891]119879 (19)
where 119906119904 V119904 119908119904 is the design coordinate of 119904119905ℎ node1199081199041015840 isthe coordinate after deformation along the direction of focalaxis and 119891 is the length of the focal axis
As shown in Figure 4 after obtaining the best fittingparaboloid the pointing error in azimuth direction causedby the lateral displacement of the vertex 119889119906 the rotationangle of the focal axis 120601V the lateral displacement of the feed119889119887 the lateral displacement of the subreflector 119889119899 and therotation angle of the subreflector 120579119904 could be obtained usingthe following equations
The distance between B1015840 and the focal axis of the subre-flector is 1198891199041 which leads to a lateral displacement 1198891199042
1198891199041 = cos 120579119904 1003816100381610038161003816119889119887 + 119889119899 + tan 12057911990411987111003816100381610038161003816 (20)
1198891199042 = 119889119904111987131198711 (21)
where 1198711 is the distance between points B and N and 1198713is the distance between points N1015840 and F119870119908 is the beam deviation factor which is related to theratio of the paraboloid focal distance to its open diameter andaperture-field distribution By omitting the vertex displace-ment of the paraboloid along the W axis the actual pointingerror caused by structural deformation can be expressed as120579119899
119889119900 = 1003816100381610038161003816tan 120601V (1198711 + 1198713 cos 120579119904 minus 1198891199042 sin 120579119904) minus (119889119899 + 1198713 sin 120579119904 + 1198891199042 cos 120579119904) + tan 120601V1198712 minus 1198891199061003816100381610038161003816radic1 + (tan 120601V)2 (22)
120579119889 = 120601V minus 119889119900119870119908119891 (23)
4 Correction Method of Estimatingthe Pointing Error
During the estimation of the pointing error 120579119899 it ishard to accurately obtain the input of the environment
loads because of the complexity of wind field Mean-while taking the modeling error of dynamic model intoaccount it usually leads to some unavoidable estimationerror of the pointing error As a result 120579119899 needs to becorrected
Shock and Vibration 5
Design paraboloid
Fitting paraboloid
U
do
휃d
B
db ds1
B㰀
휃u
휃s
dn
N
휙
F㰀
F
du
ds2
W
Figure 4 The pointing error caused by reflector deformation
From (6) the node displacement could be obtained bythe superposition of displacement of each mode so thedisplacement of 119897119905ℎ node could be expressed as follows
119904119897 = 119899sum119894=1
119862119900Φ119902119894 = 119899sum119894=1
119910119897119894 (24)
where 119910119897119894 is the displacement of the 119894119905ℎ mode So thepointing error could also be expressed as follows
120579119889 = 119899sum119894=1
120579119894 (25)
where 120579119894 is the pointing error of 119894119905ℎmode Assuming theactual pointing error 1205791198891015840 is obtained by
1205791198891015840 = 119899sum119894=1
(1 + 120572119894) 120579119894 (26)
where 120572119894 is correction weight factor of each mode Theactual displacement 1199041198971015840 of 119897119905ℎ node could be obtained
1199041198971015840 = 119899sum119894=1
(1 + 120572119894120574119894 )119910119897119894 (27)
where 120574119894 is the gain from node displacement to pointingerror of 119894119905ℎmode it could be defined as follows
120574119894 = 120579119894119910119897119894 = 120579119894119906 times 119906119910119897119894 = 100381710038171003817100381711989211991111989410038171003817100381710038172 times 1100381710038171003817100381711989211989711989410038171003817100381710038172 (28)
119892119911119894 is the transfer function from environment loadsto pointing error and 119892119897119894 is the transfer function fromenvironment loads to node displacement as shown in(15) sdot 2 denotes the H2 norm of the transfer func-tion
To simplify the pointing error expression omitting theeffect of the high-order term as (29) 119892119911119894 could be obtainedfrom (30)
1198891199042120579119904 asymp 0 (29)
119892119911119894 = (Aminus151119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)minus 1003816100381610038161003816100381610038161003816100381610038161003816(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)(1198711 + 1198713) minus (119892119894119861+ 1198713119892119894120579 + 1198713 1003816100381610038161003816119892119894119873 + 119892119894119861 + 119871111989211989412057910038161003816100381610038161198711 )+ 1198712(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894) minus (Aminus111119899sum119897=1
119892119897119894119906119904+ Aminus112
119899sum119897=1
119892119897119894V119904 + Aminus113119899sum119897=1
119892119897119894119908119904 + Aminus114119899sum119897=1
119892119897119894)1003816100381610038161003816100381610038161003816100381610038161003816sdot 119870119908119891
(30)
where 119892119894119861 is the transfer function from the environ-ment loads to the displacement of the feed of the 119894119905ℎmode 119892119894119873 is the transfer function from the environmentloads to the displacement of the subreflector of the 119894119905ℎmode and 119892119894120579 is the transfer function from the environ-ment loads to the rotation angel of the subreflector of 119894119905ℎmode
[Aminus151 Aminus152 Aminus153 Aminus154]= [0 0 0 0 1 0]An
minus1(31)
[Aminus111 Aminus112 Aminus113 Aminus114]= [1 0 0 0 0 0]An
minus1(32)
6 Shock and Vibration
The H2 norm of each transfer function could be derivedrespectively
100381710038171003817100381711989211989711989410038171003817100381710038172 = 100381710038171003817100381711988811989811989410038171003817100381710038172 1003817100381710038171003817119887119898119894100381710038171003817100381722radic120585119894120596119894 (33)
100381710038171003817100381711989211991111989410038171003817100381710038172 asymp 100381610038161003816100381611988711989811989410038161003816100381610038162radic120585119894119908119894 (1 +(1198711 + 1198713)119870119908119891 + 119870119908119891 )
sdot radic(Aminus151)2 119899sum119904=1
(119888119898119902119894119904119906119904)2 + (Aminus152)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus153)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus154)2 119899sum119904=1
(119888119898119902119894119904)2 + 119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic1198881198981199021198941198732 + 119871321198881198981199021198941205792 + 1198701199081198713 1003816100381610038161003816119887119898119894100381610038161003816100381621198911198711radic120585119894119908119894radic1198881198981199021198941198612 + 1198881198981199021198941198732 + 119871121198881198981199021198941205792 +
119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic(Aminus111)2 119899sum
119904=1
(119888119898119902119894119904119906119904)2 + (Aminus112)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus113)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus114)2 119899sum119904=1
(119888119898119902119894119904)2
(34)
where 119888119898119902119894119861 is the 119894119905ℎmode of node B which is the centralpoint of the feed 119888119898119902119894119873 is the 119894119905ℎ mode of the modal outputmatrix of nodeNwhich is the central point of the subreflectorand 119888119898119902119894120579 is the 119894119905ℎmode of the modal output matrix which isfor deriving the rotation angle of the subreflector
When the node displacements 1199041 1199042 119904119897 of the lim-ited sampling points are obtained to obtain the optimalcorrection weight factor the sum 120575119904 given in the followingshould have its smallest value And the actual pointingerror 1205791198891015840 could be derived after the correction weight factordetermined
120575119904 = radicsum119897119894=1 (119904119894 minus 1199041198941015840)2
119897 (35)
5 Model Application
Taking the antenna with a diameter of 73 meters as anexample the antenna and its finite element model are shownin Figure 5 The finite element model is used to derive modalinformation (modal matrices) of the structure which lays afoundation for the dynamic model What is more some testsare used to prove that the finite element model of the antennawas compliantwith a real antenna and some analytical resultsindicate that the proposed model is effective Next it isexplained in detail
51 Experimental Testing In order to verify the accuracy ofthe finite element model a frequency test experiment and aload deformation experiment are made to the antenna
A load deformation experiment is applied by the 100Kgimpulse in lateral loading and vertical loading respectivelyAs shown in Figure 6 red arrows represent load actiondirection Deformation testing equipment is the laser tracker
(API) An example of field testing is shown in Figure 7Numeral 1 represents the API laser tester Numeral 2 repre-sents the target location Numeral 3 represents 100Kg loadThree sets of testing were done at elevation angles of 30∘ 50∘and 70∘ respectively Comparing the ANSYS finite elementsimulation results and API test results the static maximumdeformation and the pointing error in the deformationprocess are shown in Table 1 The relative error representsthe ratio of the absolute pointing error and the test results ofpointing error
Take the dynamic process into account taking elevationangle of 70∘ for example API test results and simulationresults are compared as shown in Figure 8
It can be seen from the comparison of the results thatstatic load deformation error is less than 20 and dynamicresponse characteristics are basically consistent
Frequency test experiments were performed in two direc-tions by loading impulse loads (to generate oscillations ofthe antenna structure in the two directions) The oscillationdirections corresponding to orientation and elevation wereadopted and accelerometer sensors were placed on the twopoints with the largest amplitudes Then by using the modalanalyzer the natural frequencies in the two directions wereacquired by analyzing the data collected by the accelerometersensors and comparing the results with those of the ANSYSmodel with the same oscillation modal (with the sameoscillation) Table 2 shows a comparison of the test results andsimulation results
From the results of frequency test experiment and loaddeformation experiment it is considered that the finiteelement analysis results basically meet the accuracy require-ment The finite element model can be used as a basis fordynamic modeling and reference for real-testing structuraldeformation
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
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Shock and Vibration 3
each node of the reflector system the antenna error causedby the deformation of the structure can be syntheticallyestimated
The dynamic model of antenna structure under general-ized coordinates can be expressed as follows [15]
Mq +Dq + Kq = B0u (1)
y = Coq (2)
where the diagonal matrices M K and D are the massmatrix stiffness matrix and damping matrix respectivelyThe environment load inputs of the flexible model is 119906 Theoutput vector y is obtained by the nodal displacement vectorq B0 is the input matrix and C0 is the output matrix
To express the structure in modal coordinates the modaldisplacement is introduced which satisfies the followingequations
q = Φqm (3)
Φ is the modal shape matrix consisting of the 119895119905ℎ nodedisplacement of the 119894119905ℎmode 120601119894119895
Φ =[[[[[[[[[[[[
12060111 12060121 sdot sdot sdot 120601n112060112 12060122 sdot sdot sdot 120601n2sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1206011k 1206012k sdot sdot sdot 120601nksdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1206011nd1206012nd
sdot sdot sdot 120601nnd
]]]]]]]]]]]]
(4)
Introducing (3) into (1) and (2) and multiplying by ΦTthe dynamic equation based on modal coordinate could beobtained
Mmqm + Dmqm + Kmqm = ΦTB0u (5)
y = CoΦqm (6)
The mass matrix stiffness matrix and damping matrix weretransformed into modal mass matrix Mm modal stiffnessmatrix Km and modal damping matrix Dm respectively
Mm = ΦTMΦ (7)
Km = ΦTKΦ (8)
Dm = ΦTDΦ (9)
Meanwhile the input matrix and output matrix are trans-formed into modal input matrix Bm and modal outputmatrix Cm
ΦTB0 = Bm = [[[[[[
11988711989811198871198982sdot sdot sdot119887119898119894]]]]]]
(10)
CoΦ = Cm = [1198881198981 1198881198982 sdot sdot sdot 119888119898119894] (11)
Design main reflector
Deformed main reflector
OSub-reflector
B N
U
Wdb dn
B㰀 Feed
N㰀 휃s
Figure 3 The factors influencing the performances of antennapointing
Themodal mass matrix andmodal stiffness matrix satisfythe following equation where Ω is the natural frequencymatrix and 120596119894 is the frequency of 119894119905ℎ order
Ω2 = Mmminus1Km (12)
Ω = [[[[[[
1205961 0 sdot sdot sdot 00 1205962 sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot 120596n]]]]]] (13)
The modal damping ratio matrix Z could be expressedas (14) and the transfer function of the 119894119905ℎmode is obtainedfrom (15)
Z = [[[[[[
1205851 0 sdot sdot sdot 00 1205852 sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot 120585n]]]]]]= 05Mm
minus1DmΩminus1 (14)
119892119897119894 (120596) = 1198881198981198941198871198981198941205961198942 minus 1205962 + 2119895120585119894120596119894120596 (15)
3 The Analysis of Pointing Error Caused byStructure Deformation
To dual reflector antenna the factors influencing the per-formances of antenna pointing include the deformation ofthe main reflector the lateral displacement (perpendicular tothe focal axis) of the feed the lateral displacement and therotation of the subreflector as shown in Figure 3
4 Shock and Vibration
When analyzing the influence of the deformed mainreflector on pointing error the best fitting paraboloid shouldbe obtained based on the displacement of each node ofthe reflector There are six key parameters to confirmthe best fitting paraboloid They are the displacement ofthe vertex of the fitted parabolic reflector 119889119906 119889V 119889119908 therotation angles of the focal axis of the fitted parabolic
reflector 120601119906 120601V and the variation in the focal length119889119891The relationship among these parameters and the dis-
placement of each node could be expressed as follows [16]
119860119899120573 = 119867119899 (16)
where
119860119899 =
[[[[[[[[[[[[[[[[
119899sum119904=1
11990611990422119891119899sum119904=1
119906119904V1199042119891 minus 119899sum119904=1
119906119904 minus 119899sum119904=1
119906119904V119904 119899sum119904=1
1199061199042 119899sum119904=1
119906119904119908119904119891119899sum119904=1
119906119904V1199042119891119899sum119904=1
V1199042
2119891 minus 119899sum119904=1
V119904 minus 119899sum119904=1
V1199042119899sum119904=1
119906119904V119904 119899sum119904=1
V119904119908119904119891119899sum119904=1
1199061199041199081199042119891119899sum119904=1
V1199041199081199042119891 minus 119899sum119904=1
119908119904 minus 119899sum119904=1
V119904119908119904 119899sum119904=1
119906119904119908119904 119899sum119904=1
1199081199042119891119899sum119904=1
1199061199042119891119899sum119904=1
V1199042119891 minus119899 minus 119899sum119904=1
V119904119899sum119904=1
119906119904 119899sum119904=1
119908119904119891
]]]]]]]]]]]]]]]]
(17)
119867119899 = [ 119899sum119904=1(119908119904 minus 1199081199041015840) 119906119904 119899sum
119904=1(119908119904 minus 1199081199041015840) V119904 119899sum
119904=1(119908119904 minus 1199081199041015840)119908119904 119899sum
119904=1(119908119904 minus 1199081199041015840)]119879 (18)
120573 = [119889119906 119889V 119889119908 120601119906 120601V 119889119891]119879 (19)
where 119906119904 V119904 119908119904 is the design coordinate of 119904119905ℎ node1199081199041015840 isthe coordinate after deformation along the direction of focalaxis and 119891 is the length of the focal axis
As shown in Figure 4 after obtaining the best fittingparaboloid the pointing error in azimuth direction causedby the lateral displacement of the vertex 119889119906 the rotationangle of the focal axis 120601V the lateral displacement of the feed119889119887 the lateral displacement of the subreflector 119889119899 and therotation angle of the subreflector 120579119904 could be obtained usingthe following equations
The distance between B1015840 and the focal axis of the subre-flector is 1198891199041 which leads to a lateral displacement 1198891199042
1198891199041 = cos 120579119904 1003816100381610038161003816119889119887 + 119889119899 + tan 12057911990411987111003816100381610038161003816 (20)
1198891199042 = 119889119904111987131198711 (21)
where 1198711 is the distance between points B and N and 1198713is the distance between points N1015840 and F119870119908 is the beam deviation factor which is related to theratio of the paraboloid focal distance to its open diameter andaperture-field distribution By omitting the vertex displace-ment of the paraboloid along the W axis the actual pointingerror caused by structural deformation can be expressed as120579119899
119889119900 = 1003816100381610038161003816tan 120601V (1198711 + 1198713 cos 120579119904 minus 1198891199042 sin 120579119904) minus (119889119899 + 1198713 sin 120579119904 + 1198891199042 cos 120579119904) + tan 120601V1198712 minus 1198891199061003816100381610038161003816radic1 + (tan 120601V)2 (22)
120579119889 = 120601V minus 119889119900119870119908119891 (23)
4 Correction Method of Estimatingthe Pointing Error
During the estimation of the pointing error 120579119899 it ishard to accurately obtain the input of the environment
loads because of the complexity of wind field Mean-while taking the modeling error of dynamic model intoaccount it usually leads to some unavoidable estimationerror of the pointing error As a result 120579119899 needs to becorrected
Shock and Vibration 5
Design paraboloid
Fitting paraboloid
U
do
휃d
B
db ds1
B㰀
휃u
휃s
dn
N
휙
F㰀
F
du
ds2
W
Figure 4 The pointing error caused by reflector deformation
From (6) the node displacement could be obtained bythe superposition of displacement of each mode so thedisplacement of 119897119905ℎ node could be expressed as follows
119904119897 = 119899sum119894=1
119862119900Φ119902119894 = 119899sum119894=1
119910119897119894 (24)
where 119910119897119894 is the displacement of the 119894119905ℎ mode So thepointing error could also be expressed as follows
120579119889 = 119899sum119894=1
120579119894 (25)
where 120579119894 is the pointing error of 119894119905ℎmode Assuming theactual pointing error 1205791198891015840 is obtained by
1205791198891015840 = 119899sum119894=1
(1 + 120572119894) 120579119894 (26)
where 120572119894 is correction weight factor of each mode Theactual displacement 1199041198971015840 of 119897119905ℎ node could be obtained
1199041198971015840 = 119899sum119894=1
(1 + 120572119894120574119894 )119910119897119894 (27)
where 120574119894 is the gain from node displacement to pointingerror of 119894119905ℎmode it could be defined as follows
120574119894 = 120579119894119910119897119894 = 120579119894119906 times 119906119910119897119894 = 100381710038171003817100381711989211991111989410038171003817100381710038172 times 1100381710038171003817100381711989211989711989410038171003817100381710038172 (28)
119892119911119894 is the transfer function from environment loadsto pointing error and 119892119897119894 is the transfer function fromenvironment loads to node displacement as shown in(15) sdot 2 denotes the H2 norm of the transfer func-tion
To simplify the pointing error expression omitting theeffect of the high-order term as (29) 119892119911119894 could be obtainedfrom (30)
1198891199042120579119904 asymp 0 (29)
119892119911119894 = (Aminus151119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)minus 1003816100381610038161003816100381610038161003816100381610038161003816(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)(1198711 + 1198713) minus (119892119894119861+ 1198713119892119894120579 + 1198713 1003816100381610038161003816119892119894119873 + 119892119894119861 + 119871111989211989412057910038161003816100381610038161198711 )+ 1198712(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894) minus (Aminus111119899sum119897=1
119892119897119894119906119904+ Aminus112
119899sum119897=1
119892119897119894V119904 + Aminus113119899sum119897=1
119892119897119894119908119904 + Aminus114119899sum119897=1
119892119897119894)1003816100381610038161003816100381610038161003816100381610038161003816sdot 119870119908119891
(30)
where 119892119894119861 is the transfer function from the environ-ment loads to the displacement of the feed of the 119894119905ℎmode 119892119894119873 is the transfer function from the environmentloads to the displacement of the subreflector of the 119894119905ℎmode and 119892119894120579 is the transfer function from the environ-ment loads to the rotation angel of the subreflector of 119894119905ℎmode
[Aminus151 Aminus152 Aminus153 Aminus154]= [0 0 0 0 1 0]An
minus1(31)
[Aminus111 Aminus112 Aminus113 Aminus114]= [1 0 0 0 0 0]An
minus1(32)
6 Shock and Vibration
The H2 norm of each transfer function could be derivedrespectively
100381710038171003817100381711989211989711989410038171003817100381710038172 = 100381710038171003817100381711988811989811989410038171003817100381710038172 1003817100381710038171003817119887119898119894100381710038171003817100381722radic120585119894120596119894 (33)
100381710038171003817100381711989211991111989410038171003817100381710038172 asymp 100381610038161003816100381611988711989811989410038161003816100381610038162radic120585119894119908119894 (1 +(1198711 + 1198713)119870119908119891 + 119870119908119891 )
sdot radic(Aminus151)2 119899sum119904=1
(119888119898119902119894119904119906119904)2 + (Aminus152)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus153)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus154)2 119899sum119904=1
(119888119898119902119894119904)2 + 119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic1198881198981199021198941198732 + 119871321198881198981199021198941205792 + 1198701199081198713 1003816100381610038161003816119887119898119894100381610038161003816100381621198911198711radic120585119894119908119894radic1198881198981199021198941198612 + 1198881198981199021198941198732 + 119871121198881198981199021198941205792 +
119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic(Aminus111)2 119899sum
119904=1
(119888119898119902119894119904119906119904)2 + (Aminus112)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus113)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus114)2 119899sum119904=1
(119888119898119902119894119904)2
(34)
where 119888119898119902119894119861 is the 119894119905ℎmode of node B which is the centralpoint of the feed 119888119898119902119894119873 is the 119894119905ℎ mode of the modal outputmatrix of nodeNwhich is the central point of the subreflectorand 119888119898119902119894120579 is the 119894119905ℎmode of the modal output matrix which isfor deriving the rotation angle of the subreflector
When the node displacements 1199041 1199042 119904119897 of the lim-ited sampling points are obtained to obtain the optimalcorrection weight factor the sum 120575119904 given in the followingshould have its smallest value And the actual pointingerror 1205791198891015840 could be derived after the correction weight factordetermined
120575119904 = radicsum119897119894=1 (119904119894 minus 1199041198941015840)2
119897 (35)
5 Model Application
Taking the antenna with a diameter of 73 meters as anexample the antenna and its finite element model are shownin Figure 5 The finite element model is used to derive modalinformation (modal matrices) of the structure which lays afoundation for the dynamic model What is more some testsare used to prove that the finite element model of the antennawas compliantwith a real antenna and some analytical resultsindicate that the proposed model is effective Next it isexplained in detail
51 Experimental Testing In order to verify the accuracy ofthe finite element model a frequency test experiment and aload deformation experiment are made to the antenna
A load deformation experiment is applied by the 100Kgimpulse in lateral loading and vertical loading respectivelyAs shown in Figure 6 red arrows represent load actiondirection Deformation testing equipment is the laser tracker
(API) An example of field testing is shown in Figure 7Numeral 1 represents the API laser tester Numeral 2 repre-sents the target location Numeral 3 represents 100Kg loadThree sets of testing were done at elevation angles of 30∘ 50∘and 70∘ respectively Comparing the ANSYS finite elementsimulation results and API test results the static maximumdeformation and the pointing error in the deformationprocess are shown in Table 1 The relative error representsthe ratio of the absolute pointing error and the test results ofpointing error
Take the dynamic process into account taking elevationangle of 70∘ for example API test results and simulationresults are compared as shown in Figure 8
It can be seen from the comparison of the results thatstatic load deformation error is less than 20 and dynamicresponse characteristics are basically consistent
Frequency test experiments were performed in two direc-tions by loading impulse loads (to generate oscillations ofthe antenna structure in the two directions) The oscillationdirections corresponding to orientation and elevation wereadopted and accelerometer sensors were placed on the twopoints with the largest amplitudes Then by using the modalanalyzer the natural frequencies in the two directions wereacquired by analyzing the data collected by the accelerometersensors and comparing the results with those of the ANSYSmodel with the same oscillation modal (with the sameoscillation) Table 2 shows a comparison of the test results andsimulation results
From the results of frequency test experiment and loaddeformation experiment it is considered that the finiteelement analysis results basically meet the accuracy require-ment The finite element model can be used as a basis fordynamic modeling and reference for real-testing structuraldeformation
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
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4 Shock and Vibration
When analyzing the influence of the deformed mainreflector on pointing error the best fitting paraboloid shouldbe obtained based on the displacement of each node ofthe reflector There are six key parameters to confirmthe best fitting paraboloid They are the displacement ofthe vertex of the fitted parabolic reflector 119889119906 119889V 119889119908 therotation angles of the focal axis of the fitted parabolic
reflector 120601119906 120601V and the variation in the focal length119889119891The relationship among these parameters and the dis-
placement of each node could be expressed as follows [16]
119860119899120573 = 119867119899 (16)
where
119860119899 =
[[[[[[[[[[[[[[[[
119899sum119904=1
11990611990422119891119899sum119904=1
119906119904V1199042119891 minus 119899sum119904=1
119906119904 minus 119899sum119904=1
119906119904V119904 119899sum119904=1
1199061199042 119899sum119904=1
119906119904119908119904119891119899sum119904=1
119906119904V1199042119891119899sum119904=1
V1199042
2119891 minus 119899sum119904=1
V119904 minus 119899sum119904=1
V1199042119899sum119904=1
119906119904V119904 119899sum119904=1
V119904119908119904119891119899sum119904=1
1199061199041199081199042119891119899sum119904=1
V1199041199081199042119891 minus 119899sum119904=1
119908119904 minus 119899sum119904=1
V119904119908119904 119899sum119904=1
119906119904119908119904 119899sum119904=1
1199081199042119891119899sum119904=1
1199061199042119891119899sum119904=1
V1199042119891 minus119899 minus 119899sum119904=1
V119904119899sum119904=1
119906119904 119899sum119904=1
119908119904119891
]]]]]]]]]]]]]]]]
(17)
119867119899 = [ 119899sum119904=1(119908119904 minus 1199081199041015840) 119906119904 119899sum
119904=1(119908119904 minus 1199081199041015840) V119904 119899sum
119904=1(119908119904 minus 1199081199041015840)119908119904 119899sum
119904=1(119908119904 minus 1199081199041015840)]119879 (18)
120573 = [119889119906 119889V 119889119908 120601119906 120601V 119889119891]119879 (19)
where 119906119904 V119904 119908119904 is the design coordinate of 119904119905ℎ node1199081199041015840 isthe coordinate after deformation along the direction of focalaxis and 119891 is the length of the focal axis
As shown in Figure 4 after obtaining the best fittingparaboloid the pointing error in azimuth direction causedby the lateral displacement of the vertex 119889119906 the rotationangle of the focal axis 120601V the lateral displacement of the feed119889119887 the lateral displacement of the subreflector 119889119899 and therotation angle of the subreflector 120579119904 could be obtained usingthe following equations
The distance between B1015840 and the focal axis of the subre-flector is 1198891199041 which leads to a lateral displacement 1198891199042
1198891199041 = cos 120579119904 1003816100381610038161003816119889119887 + 119889119899 + tan 12057911990411987111003816100381610038161003816 (20)
1198891199042 = 119889119904111987131198711 (21)
where 1198711 is the distance between points B and N and 1198713is the distance between points N1015840 and F119870119908 is the beam deviation factor which is related to theratio of the paraboloid focal distance to its open diameter andaperture-field distribution By omitting the vertex displace-ment of the paraboloid along the W axis the actual pointingerror caused by structural deformation can be expressed as120579119899
119889119900 = 1003816100381610038161003816tan 120601V (1198711 + 1198713 cos 120579119904 minus 1198891199042 sin 120579119904) minus (119889119899 + 1198713 sin 120579119904 + 1198891199042 cos 120579119904) + tan 120601V1198712 minus 1198891199061003816100381610038161003816radic1 + (tan 120601V)2 (22)
120579119889 = 120601V minus 119889119900119870119908119891 (23)
4 Correction Method of Estimatingthe Pointing Error
During the estimation of the pointing error 120579119899 it ishard to accurately obtain the input of the environment
loads because of the complexity of wind field Mean-while taking the modeling error of dynamic model intoaccount it usually leads to some unavoidable estimationerror of the pointing error As a result 120579119899 needs to becorrected
Shock and Vibration 5
Design paraboloid
Fitting paraboloid
U
do
휃d
B
db ds1
B㰀
휃u
휃s
dn
N
휙
F㰀
F
du
ds2
W
Figure 4 The pointing error caused by reflector deformation
From (6) the node displacement could be obtained bythe superposition of displacement of each mode so thedisplacement of 119897119905ℎ node could be expressed as follows
119904119897 = 119899sum119894=1
119862119900Φ119902119894 = 119899sum119894=1
119910119897119894 (24)
where 119910119897119894 is the displacement of the 119894119905ℎ mode So thepointing error could also be expressed as follows
120579119889 = 119899sum119894=1
120579119894 (25)
where 120579119894 is the pointing error of 119894119905ℎmode Assuming theactual pointing error 1205791198891015840 is obtained by
1205791198891015840 = 119899sum119894=1
(1 + 120572119894) 120579119894 (26)
where 120572119894 is correction weight factor of each mode Theactual displacement 1199041198971015840 of 119897119905ℎ node could be obtained
1199041198971015840 = 119899sum119894=1
(1 + 120572119894120574119894 )119910119897119894 (27)
where 120574119894 is the gain from node displacement to pointingerror of 119894119905ℎmode it could be defined as follows
120574119894 = 120579119894119910119897119894 = 120579119894119906 times 119906119910119897119894 = 100381710038171003817100381711989211991111989410038171003817100381710038172 times 1100381710038171003817100381711989211989711989410038171003817100381710038172 (28)
119892119911119894 is the transfer function from environment loadsto pointing error and 119892119897119894 is the transfer function fromenvironment loads to node displacement as shown in(15) sdot 2 denotes the H2 norm of the transfer func-tion
To simplify the pointing error expression omitting theeffect of the high-order term as (29) 119892119911119894 could be obtainedfrom (30)
1198891199042120579119904 asymp 0 (29)
119892119911119894 = (Aminus151119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)minus 1003816100381610038161003816100381610038161003816100381610038161003816(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)(1198711 + 1198713) minus (119892119894119861+ 1198713119892119894120579 + 1198713 1003816100381610038161003816119892119894119873 + 119892119894119861 + 119871111989211989412057910038161003816100381610038161198711 )+ 1198712(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894) minus (Aminus111119899sum119897=1
119892119897119894119906119904+ Aminus112
119899sum119897=1
119892119897119894V119904 + Aminus113119899sum119897=1
119892119897119894119908119904 + Aminus114119899sum119897=1
119892119897119894)1003816100381610038161003816100381610038161003816100381610038161003816sdot 119870119908119891
(30)
where 119892119894119861 is the transfer function from the environ-ment loads to the displacement of the feed of the 119894119905ℎmode 119892119894119873 is the transfer function from the environmentloads to the displacement of the subreflector of the 119894119905ℎmode and 119892119894120579 is the transfer function from the environ-ment loads to the rotation angel of the subreflector of 119894119905ℎmode
[Aminus151 Aminus152 Aminus153 Aminus154]= [0 0 0 0 1 0]An
minus1(31)
[Aminus111 Aminus112 Aminus113 Aminus114]= [1 0 0 0 0 0]An
minus1(32)
6 Shock and Vibration
The H2 norm of each transfer function could be derivedrespectively
100381710038171003817100381711989211989711989410038171003817100381710038172 = 100381710038171003817100381711988811989811989410038171003817100381710038172 1003817100381710038171003817119887119898119894100381710038171003817100381722radic120585119894120596119894 (33)
100381710038171003817100381711989211991111989410038171003817100381710038172 asymp 100381610038161003816100381611988711989811989410038161003816100381610038162radic120585119894119908119894 (1 +(1198711 + 1198713)119870119908119891 + 119870119908119891 )
sdot radic(Aminus151)2 119899sum119904=1
(119888119898119902119894119904119906119904)2 + (Aminus152)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus153)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus154)2 119899sum119904=1
(119888119898119902119894119904)2 + 119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic1198881198981199021198941198732 + 119871321198881198981199021198941205792 + 1198701199081198713 1003816100381610038161003816119887119898119894100381610038161003816100381621198911198711radic120585119894119908119894radic1198881198981199021198941198612 + 1198881198981199021198941198732 + 119871121198881198981199021198941205792 +
119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic(Aminus111)2 119899sum
119904=1
(119888119898119902119894119904119906119904)2 + (Aminus112)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus113)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus114)2 119899sum119904=1
(119888119898119902119894119904)2
(34)
where 119888119898119902119894119861 is the 119894119905ℎmode of node B which is the centralpoint of the feed 119888119898119902119894119873 is the 119894119905ℎ mode of the modal outputmatrix of nodeNwhich is the central point of the subreflectorand 119888119898119902119894120579 is the 119894119905ℎmode of the modal output matrix which isfor deriving the rotation angle of the subreflector
When the node displacements 1199041 1199042 119904119897 of the lim-ited sampling points are obtained to obtain the optimalcorrection weight factor the sum 120575119904 given in the followingshould have its smallest value And the actual pointingerror 1205791198891015840 could be derived after the correction weight factordetermined
120575119904 = radicsum119897119894=1 (119904119894 minus 1199041198941015840)2
119897 (35)
5 Model Application
Taking the antenna with a diameter of 73 meters as anexample the antenna and its finite element model are shownin Figure 5 The finite element model is used to derive modalinformation (modal matrices) of the structure which lays afoundation for the dynamic model What is more some testsare used to prove that the finite element model of the antennawas compliantwith a real antenna and some analytical resultsindicate that the proposed model is effective Next it isexplained in detail
51 Experimental Testing In order to verify the accuracy ofthe finite element model a frequency test experiment and aload deformation experiment are made to the antenna
A load deformation experiment is applied by the 100Kgimpulse in lateral loading and vertical loading respectivelyAs shown in Figure 6 red arrows represent load actiondirection Deformation testing equipment is the laser tracker
(API) An example of field testing is shown in Figure 7Numeral 1 represents the API laser tester Numeral 2 repre-sents the target location Numeral 3 represents 100Kg loadThree sets of testing were done at elevation angles of 30∘ 50∘and 70∘ respectively Comparing the ANSYS finite elementsimulation results and API test results the static maximumdeformation and the pointing error in the deformationprocess are shown in Table 1 The relative error representsthe ratio of the absolute pointing error and the test results ofpointing error
Take the dynamic process into account taking elevationangle of 70∘ for example API test results and simulationresults are compared as shown in Figure 8
It can be seen from the comparison of the results thatstatic load deformation error is less than 20 and dynamicresponse characteristics are basically consistent
Frequency test experiments were performed in two direc-tions by loading impulse loads (to generate oscillations ofthe antenna structure in the two directions) The oscillationdirections corresponding to orientation and elevation wereadopted and accelerometer sensors were placed on the twopoints with the largest amplitudes Then by using the modalanalyzer the natural frequencies in the two directions wereacquired by analyzing the data collected by the accelerometersensors and comparing the results with those of the ANSYSmodel with the same oscillation modal (with the sameoscillation) Table 2 shows a comparison of the test results andsimulation results
From the results of frequency test experiment and loaddeformation experiment it is considered that the finiteelement analysis results basically meet the accuracy require-ment The finite element model can be used as a basis fordynamic modeling and reference for real-testing structuraldeformation
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
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Shock and Vibration 5
Design paraboloid
Fitting paraboloid
U
do
휃d
B
db ds1
B㰀
휃u
휃s
dn
N
휙
F㰀
F
du
ds2
W
Figure 4 The pointing error caused by reflector deformation
From (6) the node displacement could be obtained bythe superposition of displacement of each mode so thedisplacement of 119897119905ℎ node could be expressed as follows
119904119897 = 119899sum119894=1
119862119900Φ119902119894 = 119899sum119894=1
119910119897119894 (24)
where 119910119897119894 is the displacement of the 119894119905ℎ mode So thepointing error could also be expressed as follows
120579119889 = 119899sum119894=1
120579119894 (25)
where 120579119894 is the pointing error of 119894119905ℎmode Assuming theactual pointing error 1205791198891015840 is obtained by
1205791198891015840 = 119899sum119894=1
(1 + 120572119894) 120579119894 (26)
where 120572119894 is correction weight factor of each mode Theactual displacement 1199041198971015840 of 119897119905ℎ node could be obtained
1199041198971015840 = 119899sum119894=1
(1 + 120572119894120574119894 )119910119897119894 (27)
where 120574119894 is the gain from node displacement to pointingerror of 119894119905ℎmode it could be defined as follows
120574119894 = 120579119894119910119897119894 = 120579119894119906 times 119906119910119897119894 = 100381710038171003817100381711989211991111989410038171003817100381710038172 times 1100381710038171003817100381711989211989711989410038171003817100381710038172 (28)
119892119911119894 is the transfer function from environment loadsto pointing error and 119892119897119894 is the transfer function fromenvironment loads to node displacement as shown in(15) sdot 2 denotes the H2 norm of the transfer func-tion
To simplify the pointing error expression omitting theeffect of the high-order term as (29) 119892119911119894 could be obtainedfrom (30)
1198891199042120579119904 asymp 0 (29)
119892119911119894 = (Aminus151119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)minus 1003816100381610038161003816100381610038161003816100381610038161003816(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894)(1198711 + 1198713) minus (119892119894119861+ 1198713119892119894120579 + 1198713 1003816100381610038161003816119892119894119873 + 119892119894119861 + 119871111989211989412057910038161003816100381610038161198711 )+ 1198712(Aminus151
119899sum119897=1
119892119897119894119906119904 + Aminus152119899sum119897=1
119892119897119894V119904+ Aminus153
119899sum119897=1
119892119897119894119908119904 + Aminus154119899sum119897=1
119892119897119894) minus (Aminus111119899sum119897=1
119892119897119894119906119904+ Aminus112
119899sum119897=1
119892119897119894V119904 + Aminus113119899sum119897=1
119892119897119894119908119904 + Aminus114119899sum119897=1
119892119897119894)1003816100381610038161003816100381610038161003816100381610038161003816sdot 119870119908119891
(30)
where 119892119894119861 is the transfer function from the environ-ment loads to the displacement of the feed of the 119894119905ℎmode 119892119894119873 is the transfer function from the environmentloads to the displacement of the subreflector of the 119894119905ℎmode and 119892119894120579 is the transfer function from the environ-ment loads to the rotation angel of the subreflector of 119894119905ℎmode
[Aminus151 Aminus152 Aminus153 Aminus154]= [0 0 0 0 1 0]An
minus1(31)
[Aminus111 Aminus112 Aminus113 Aminus114]= [1 0 0 0 0 0]An
minus1(32)
6 Shock and Vibration
The H2 norm of each transfer function could be derivedrespectively
100381710038171003817100381711989211989711989410038171003817100381710038172 = 100381710038171003817100381711988811989811989410038171003817100381710038172 1003817100381710038171003817119887119898119894100381710038171003817100381722radic120585119894120596119894 (33)
100381710038171003817100381711989211991111989410038171003817100381710038172 asymp 100381610038161003816100381611988711989811989410038161003816100381610038162radic120585119894119908119894 (1 +(1198711 + 1198713)119870119908119891 + 119870119908119891 )
sdot radic(Aminus151)2 119899sum119904=1
(119888119898119902119894119904119906119904)2 + (Aminus152)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus153)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus154)2 119899sum119904=1
(119888119898119902119894119904)2 + 119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic1198881198981199021198941198732 + 119871321198881198981199021198941205792 + 1198701199081198713 1003816100381610038161003816119887119898119894100381610038161003816100381621198911198711radic120585119894119908119894radic1198881198981199021198941198612 + 1198881198981199021198941198732 + 119871121198881198981199021198941205792 +
119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic(Aminus111)2 119899sum
119904=1
(119888119898119902119894119904119906119904)2 + (Aminus112)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus113)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus114)2 119899sum119904=1
(119888119898119902119894119904)2
(34)
where 119888119898119902119894119861 is the 119894119905ℎmode of node B which is the centralpoint of the feed 119888119898119902119894119873 is the 119894119905ℎ mode of the modal outputmatrix of nodeNwhich is the central point of the subreflectorand 119888119898119902119894120579 is the 119894119905ℎmode of the modal output matrix which isfor deriving the rotation angle of the subreflector
When the node displacements 1199041 1199042 119904119897 of the lim-ited sampling points are obtained to obtain the optimalcorrection weight factor the sum 120575119904 given in the followingshould have its smallest value And the actual pointingerror 1205791198891015840 could be derived after the correction weight factordetermined
120575119904 = radicsum119897119894=1 (119904119894 minus 1199041198941015840)2
119897 (35)
5 Model Application
Taking the antenna with a diameter of 73 meters as anexample the antenna and its finite element model are shownin Figure 5 The finite element model is used to derive modalinformation (modal matrices) of the structure which lays afoundation for the dynamic model What is more some testsare used to prove that the finite element model of the antennawas compliantwith a real antenna and some analytical resultsindicate that the proposed model is effective Next it isexplained in detail
51 Experimental Testing In order to verify the accuracy ofthe finite element model a frequency test experiment and aload deformation experiment are made to the antenna
A load deformation experiment is applied by the 100Kgimpulse in lateral loading and vertical loading respectivelyAs shown in Figure 6 red arrows represent load actiondirection Deformation testing equipment is the laser tracker
(API) An example of field testing is shown in Figure 7Numeral 1 represents the API laser tester Numeral 2 repre-sents the target location Numeral 3 represents 100Kg loadThree sets of testing were done at elevation angles of 30∘ 50∘and 70∘ respectively Comparing the ANSYS finite elementsimulation results and API test results the static maximumdeformation and the pointing error in the deformationprocess are shown in Table 1 The relative error representsthe ratio of the absolute pointing error and the test results ofpointing error
Take the dynamic process into account taking elevationangle of 70∘ for example API test results and simulationresults are compared as shown in Figure 8
It can be seen from the comparison of the results thatstatic load deformation error is less than 20 and dynamicresponse characteristics are basically consistent
Frequency test experiments were performed in two direc-tions by loading impulse loads (to generate oscillations ofthe antenna structure in the two directions) The oscillationdirections corresponding to orientation and elevation wereadopted and accelerometer sensors were placed on the twopoints with the largest amplitudes Then by using the modalanalyzer the natural frequencies in the two directions wereacquired by analyzing the data collected by the accelerometersensors and comparing the results with those of the ANSYSmodel with the same oscillation modal (with the sameoscillation) Table 2 shows a comparison of the test results andsimulation results
From the results of frequency test experiment and loaddeformation experiment it is considered that the finiteelement analysis results basically meet the accuracy require-ment The finite element model can be used as a basis fordynamic modeling and reference for real-testing structuraldeformation
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
6 Shock and Vibration
The H2 norm of each transfer function could be derivedrespectively
100381710038171003817100381711989211989711989410038171003817100381710038172 = 100381710038171003817100381711988811989811989410038171003817100381710038172 1003817100381710038171003817119887119898119894100381710038171003817100381722radic120585119894120596119894 (33)
100381710038171003817100381711989211991111989410038171003817100381710038172 asymp 100381610038161003816100381611988711989811989410038161003816100381610038162radic120585119894119908119894 (1 +(1198711 + 1198713)119870119908119891 + 119870119908119891 )
sdot radic(Aminus151)2 119899sum119904=1
(119888119898119902119894119904119906119904)2 + (Aminus152)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus153)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus154)2 119899sum119904=1
(119888119898119902119894119904)2 + 119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic1198881198981199021198941198732 + 119871321198881198981199021198941205792 + 1198701199081198713 1003816100381610038161003816119887119898119894100381610038161003816100381621198911198711radic120585119894119908119894radic1198881198981199021198941198612 + 1198881198981199021198941198732 + 119871121198881198981199021198941205792 +
119870119908 100381610038161003816100381611988711989811989410038161003816100381610038162119891radic120585119894119908119894sdot radic(Aminus111)2 119899sum
119904=1
(119888119898119902119894119904119906119904)2 + (Aminus112)2 119899sum119904=1
(119888119898119902119894119904V119904)2 + (Aminus113)2 119899sum119904=1
(119888119898119902119894119904119908119904)2 + (Aminus114)2 119899sum119904=1
(119888119898119902119894119904)2
(34)
where 119888119898119902119894119861 is the 119894119905ℎmode of node B which is the centralpoint of the feed 119888119898119902119894119873 is the 119894119905ℎ mode of the modal outputmatrix of nodeNwhich is the central point of the subreflectorand 119888119898119902119894120579 is the 119894119905ℎmode of the modal output matrix which isfor deriving the rotation angle of the subreflector
When the node displacements 1199041 1199042 119904119897 of the lim-ited sampling points are obtained to obtain the optimalcorrection weight factor the sum 120575119904 given in the followingshould have its smallest value And the actual pointingerror 1205791198891015840 could be derived after the correction weight factordetermined
120575119904 = radicsum119897119894=1 (119904119894 minus 1199041198941015840)2
119897 (35)
5 Model Application
Taking the antenna with a diameter of 73 meters as anexample the antenna and its finite element model are shownin Figure 5 The finite element model is used to derive modalinformation (modal matrices) of the structure which lays afoundation for the dynamic model What is more some testsare used to prove that the finite element model of the antennawas compliantwith a real antenna and some analytical resultsindicate that the proposed model is effective Next it isexplained in detail
51 Experimental Testing In order to verify the accuracy ofthe finite element model a frequency test experiment and aload deformation experiment are made to the antenna
A load deformation experiment is applied by the 100Kgimpulse in lateral loading and vertical loading respectivelyAs shown in Figure 6 red arrows represent load actiondirection Deformation testing equipment is the laser tracker
(API) An example of field testing is shown in Figure 7Numeral 1 represents the API laser tester Numeral 2 repre-sents the target location Numeral 3 represents 100Kg loadThree sets of testing were done at elevation angles of 30∘ 50∘and 70∘ respectively Comparing the ANSYS finite elementsimulation results and API test results the static maximumdeformation and the pointing error in the deformationprocess are shown in Table 1 The relative error representsthe ratio of the absolute pointing error and the test results ofpointing error
Take the dynamic process into account taking elevationangle of 70∘ for example API test results and simulationresults are compared as shown in Figure 8
It can be seen from the comparison of the results thatstatic load deformation error is less than 20 and dynamicresponse characteristics are basically consistent
Frequency test experiments were performed in two direc-tions by loading impulse loads (to generate oscillations ofthe antenna structure in the two directions) The oscillationdirections corresponding to orientation and elevation wereadopted and accelerometer sensors were placed on the twopoints with the largest amplitudes Then by using the modalanalyzer the natural frequencies in the two directions wereacquired by analyzing the data collected by the accelerometersensors and comparing the results with those of the ANSYSmodel with the same oscillation modal (with the sameoscillation) Table 2 shows a comparison of the test results andsimulation results
From the results of frequency test experiment and loaddeformation experiment it is considered that the finiteelement analysis results basically meet the accuracy require-ment The finite element model can be used as a basis fordynamic modeling and reference for real-testing structuraldeformation
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 7
Elevation angle encoder
Azimuth angle encoder
Figure 5 The real and finite element model of 73m antenna
Figure 6 The position and direction of the applied load
52 Numerical Calculation For pointing error simulationanalysis taking a random disturbance as an example theaverage wind speed is 10m s according to the spectrumcharacteristics of wind disturbance The equivalent windforce acting laterally on the 73m antenna reflector is shownin Figure 9
The maximum pointing error is about 00041∘ whichcaused by the structural deformation under the wind dis-turbance by applying the pointing error analysis modeldescribed in this article see Figure 10
Because the wind disturbance modeling error is not con-sidered in the above-mentioned pointing error estimationthe correction method described in the article will be usedto correct the pointing error
Suppose the reflector system uniformly distributes 15deformation sampling points see Figure 11
When the wind disturbance modeling error is not intro-duced the node deformation is calculated from the dynamicmodel described in this paper see Figure 12 Due to the sym-metry of the reflector system and the loading action it onlyselects number 1 2 5 9 10 and 14 node displacements forcomparison x1 x2 x5 x9 x10 and x14 are the correspondingnode deformation respectively
The finite element software analysis results replace theactual deformation information in pointing error correctionIn the process of the finite element analysis the loadmodelingerror is introduced by the transient analysis of the ANSYSsoftware the deformation of the nodes (s1 s2 s5 s9 s10 ands14) is shown in Figure 13
1
23
Figure 7 An example of field testing
53 Analytical Results Thedeformation information of thesenodes is regarded as the measured deformation informationThe correction method proposed in this paper is used tocorrect the pointing error shown in Figure 10 Comparing thecorrection result of pointing error before and after it can beseen that the maximum value of the pointing error is 00041∘which is obtained by relying solely on the pointing erroranalysis model as shown in Figure 14 After considering thewind disturbance model error the corrected pointing error is00054∘
According to the correction of estimating the pointingerror caused by the structural deformation a controller isdesigned as shown in Figure 15 (PID controller is the mostcommon techniques in engineering projects)
Under a wind with the speed of 10 ms Figure 16 showsthe pointing performances of the antenna servo system withPID controller Before the compensation of the pointingerror caused by wind pressure the maximum pointing errorwould be up to 00054∘ But the maximum pointing error isonly about 00008∘ by introducing 1205791198891015840 into controller of theposition loop Therefore it is necessary to propose a methodto estimate the pointing error caused by wind accurately
6 Conclusion
The influence of the large reflector antenna flexible deforma-tion on the pointing accuracy has become more and moresignificant The traditional pointing error estimation methodeither depends on finite element software analysis and thuscannot be directly applied to pointing control or depends onthe modeling accuracy of the load model and the pointingerror analysis model and thus cannot guarantee the accuracyof the error estimation Against these problems this paperproposes a correction method of pointing error caused bystructural deformation based on optimization of correctionweight factor of each order of pointing error Through theexperiment and simulation of the 73m antenna it showsthat the method can effectively correct the pointing errorestimation by using the measured deformation consequently
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
8 Shock and Vibration
Table1Th
ecom
paris
onof
static
maxim
umdeform
ationandthep
ointingerror
Loading
direction
ELangle
Simulationresults
ofdeform
ation
(mm)
Simulationresults
ofpo
intin
gerror
()
Testresults
ofdeform
ation
(mm)
Testresults
ofpo
intin
gerror
()
Relativ
eerror
()
Lateral
direction
70080
288
068
241
1950
50066
180
055
152
1842
30041
113
050
136
1691
Vertical
direction
7013
0432
123
411
511
5014
5504
144
489
307
3016
3612
152
630
444
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 9
TestSimulation
1 2 3 4 5 6 70Time(s)
minus1minus08minus06minus04minus02
002040608
1D
ispla
cem
ent (
mm
)
(a) The dynamic oscillatory in time domain
minus02minus015minus01minus005
0005
01015
Erro
r(m
m)
1 2 3 4 5 6 70Time(s)
(b) The error between test and simulation
Figure 8 The comparison of the dynamic oscillatory of the sampling point
2 4 6 80 12 14 16 18 2010
Time(s)
0100020003000400050006000
Torq
ue(N
m)
Figure 9 The equivalent wind force acting laterally on reflector
minus050
051
152
253
354
45
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 10 The pointing error caused by deformation of the reflector
Main reflector
Subreflector
Feed
Sampling point
1
2
3
4
5
6 7
810
11
12
139 14
15
Figure 11 The distribution of sampling points on the reflector
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
10 Shock and Vibration
Table 2 Natural frequency measurement tests
Test content Load (kg) Test result (Hz) Simulation result (Hz) Relative error ()Rotation mode in EL 100 684 7625 1148Rotation mode in EL 50 684 7625 1148Rotation mode in AZ 100 2734 26893 163Rotation mode in AZ 50 2734 26893 163
x1x2x5
x9x10x14
2 4 6 80 12 14 16 18 2010
Time(s)
005
115
225
335
445
Nod
e disp
lace
men
t(mm
)
Figure 12 The nodal displacement obtained from dynamic model
s1s2s5
s9s10s14
005
115
225
335
Nod
e disp
lace
men
t(mm
)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 13 The nodal displacement with considering the load modeling error
minus10123456
Poin
ting
erro
r(de
g10
00)
Calculated errorModified error
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 14 The comparison of pointing error before and after the correction
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 11
Velocitycontroller Drivers Antenna
Rigid modelPID
controller+
-
+ PointingRef
Plant
-
Wind pressure
Dynamic model
Analysis of pointing error
+
+
Error correction
휃d
휃㰀
휃
휃㰀d
Figure 15 Antenna servo system
Before compensationAer compensation
minus10123456
Poin
ting
erro
r(de
g10
00)
2 4 6 80 12 14 16 18 2010
Time(s)
Figure 16 The pointing performance after compensation
to improve the accuracy of the pointing error after compen-sation with PID controller
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National 973 Programunder Grant no 2015CB857100 the National Natural ScienceFoundation of China under Grant nos 51705387 51575419and 51490660 the National 111 Project under Grant noB14042 the China Postdoctoral Science Foundation underGrant no 2017M613078 the Fundamental Research Funds forthe Central Universities under Grant no JBX170414 and theCAS ldquoLight of West Chinardquo Program under Grant nos 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023
References
[1] J Zhang J Huang R Song and L Qiu ldquoAnalysis of reflectorvibration-induced pointing errors for large antennas subject towind disturbancerdquo IEEE Antennas and Propagation Magazinevol 57 no 6 pp 46ndash61 2015
[2] M Brenner M J Britcliffe and W A Imbriale ldquoGravitydeformation measurements of 70 m reflector surfacesrdquo IEEEAntennas and Propagation Magazine vol 44 no 6 pp 187ndash1922001
[3] R Subrahmanyan ldquoPhotogrammetric measurement of thegravity deformation in a Cassegrain antennardquo IEEE Transac-tions onAntennas and Propagation vol 53 no 8 pp 2590ndash25962005
[4] C S Wang B Y Duan and Y Y Qiu ldquoOn distorted surfaceanalysis and multidisciplinary structural optimization of largereflector antennasrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 6 pp 519ndash528 2007
[5] W Wang C Wang B Duan and G Leng ldquoCompensation forgravity deformation via subreflector motion of 65 m shapedCassegrain antennardquo Microwaves Antennas and Propagationletter vol 8 no 3 pp 158ndash164 2013
[6] S Xu Y Rahmat-Samii and W A Imbriale ldquoSubreflectarraysfor reflector surface distortion compensationrdquo IEEE Transac-tions on Antennas and Propagation vol 57 no 2 pp 364ndash3722009
[7] N Hu H Bao B Duan G Yang and Y Zong ldquoTopologyoptimization of reflector antennas based on integrated thermal-structural-electromagnetic analysisrdquo Structural and Multidisci-plinary Optimization pp 1ndash8 2016
[8] C S Wang B Y Duan F S Zhang and M B Zhu ldquoCoupledstructural-electromagnetic-thermal modelling and analysis ofactive phased array antennasrdquo IET Microwaves Antennas ampPropagation vol 4 no 2 pp 247ndash257 2010
[9] P Li B Y Duan W Wang and F Zheng ldquoElectromechanicalcoupling analysis of ground reflector antennas under solarradiationrdquo IEEE Antennas and Propagation Magazine vol 54no 5 pp 40ndash57 2012
[10] W Gawronski J A Mellstrom and B Bienkiewicz ldquoAntennamean wind torques A comparison of field and wind-tunneldatardquo IEEE Antennas and Propagation Magazine vol 47 no 5pp 55ndash59 2005
[11] W Gawronski ldquoModeling wind-gust disturbances for theanalysis of antenna pointing accuracyrdquo IEEE Antennas andPropagation Magazine vol 46 no 1 pp 50ndash58 2004
[12] B Y Duan and C S Wang ldquoReflector antenna distortionanalysis using MEFCMrdquo IEEE Transactions on Antennas andPropagation vol 57 no 10 pp 3409ndash3413 2009
[13] J Zhang J Huang J Zhou C Wang and Y Zhu ldquoA com-pensator for large antennas based on pointing error estimationunder a wind loadrdquo IEEE Transactions on Control SystemsTechnology vol 5 no 25 pp 1912ndash1920 2017
[14] J Zhang J Huang S Wang and C Wang ldquoAn active pointingcompensator for large beam waveguide antenna under winddisturbancerdquo IEEEASME Transactions onMechatronics vol 21no 2 pp 860ndash871 2016
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
12 Shock and Vibration
[15] W Gawronski ldquoAnalytical modelsrdquo in Modeling and Control ofAntennas and Telescopes pp 13ndash50 Springer New York NYUSA 2004
[16] C S Wang B Y Duan and Y Y Qiu ldquoPrecise algorithm forsurface errors of reflector antennas and analysis of its electricalperformancerdquo Chinese Journal of Radio Science vol 21 pp 23ndash28 2006
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom