STRAIN GAGE ANGULAR ACCELERATION SENSOR FACILITATES PRECISE CONTROL OF VIBRATION TABLE

Transcription

1 STRAIN GAGE ANGULAR ACCELERATION SENSOR FACILITATES PRECISE CONTROL OF VIBRATION TABLE Howard S. Havlicsek (Acutroic USA, Ic.) Preseted 14 September 1999 Symposium Gyro Techology 1999 Stuttgart, Germay

2 STRAIN GAGE ANGULAR ACCELERATION SENSOR FACILITATES PRECISE CONTROL OF VIBRATION TABLE Howard S. Havlicsek (Acutroic USA, Ic.) Abstract Key words: Agular Accelerometer,, Vibratio Table Agular vibratio tables are ofte used for product developmet ad o the productio floor to test ad characterize iertial sesors. The fidelity, stability ad frequecy respose of sythesized vibratory motio are sigificat factors that ca limit the ability to fully test iertial products. This paper offers a approach to improvig the performace of agular vibratio motio simulators by recogizig those factors that limit performace ad the employig a sesig techique that overcomes oe domiat deficiecy. This paper discusses the aalysis, desig, ad testig of a agular acceleratio sesor ad summarizes operatio ad performace i the cotext of a Acutroic rotary vibratio table. The sesor ca be icorporated ito systems of various cofiguratios depedig o mechaical costraits ad performace goals. May of the classical problems with covetioal agular accelerometers are miimized due to the symmetry ad iheret simplicity of this agular sesor ad dyamic sesor respose is completely elimiated. Theoretical ad empirical data are preseted to substatiate the aalytical models. Itroductio Agular vibratio sythesis has traditioally bee implemeted o covetioal rate tables usig either a agular accelerometer or multiple tagetially mouted liear accelerometers for feedback sesig. System performace is ofte compromised by the phase shift (badwidth) of the feedback sesor, the shaft compliace betwee the motor ad the table top, ad/or off axis aligmet sesitivities. Cotroller compesatio icludes otchig mechaical resoaces ad gai profilig filters all of which add more phase shift ad cotribute to reduced closed loop badwidth of the motio system. This paper defies a uique solutio for a agular vibratio table which uses a strai gage agular sesor to measure the agular acceleratio of the table top. The sesor cosists of a member attached at opposite eds of the output shaft such that it experieces the widup of the shaft wheever torque is trasferred from the motor to the table top. The strai gage sesor measures the shaft output torque, which is proportioal to the free body acceleratio of the table top ad load. Acceleratio measured i this maer has o performace limitig dyamics, much the same way that a perfect agular accelerometer mouted directly o the table top would sese acceleratio with ifiite badwidth ad zero phase shift. I this cofiguratio, the etire vibratio table (icludig the UUT) becomes a closed loop agular accelerometer. The advatage of this approach is that the additioal dyamic error of a realistic accelerometer required i a covetioal vibratio table is elimiated ad total ope loop phase shift is sigificatly less due to the lower order of the plat/sesor dyamics. The closed acceleratio loop ca the be adjusted for higher badwidth ad less closed loop phase shift. The cotroller implemets a fairly covetioal high badwidth ier acceleratio loop ad a low badwidth outer positio loop provides ull drift stabilizatio. I practice it is possible to achieve acceleratio badwidths that are beyod the shaft resoat frequecy. The frequecy respose of a covetioal vibratio table is historically limited to 5 Hz usig available acceleratio sesors. A classical table desig has demostrated badwidth i excess of 1 Hz with less tha 3 db deviatio i amplitude respose by icorporatig a strai gage acceleratio sesor. Whe frequecy respose requiremets are pushed to the limit may factors come ito the picture that are ormally igored. Factors such as base reactio, shaft resoace, UUT/fixture moutig, motor reactace, exteral disturbace, power amp badwidth, or frictio, could degrade the performace of a system. The desig of a vibratio table must cosider these issues to esure that their combied effect ca be maaged.

3 Desig 1. Geeral descriptio of the agular sesor i the cotext of a Acutroic vibratio table. The Acutroic series 1/AV agular vibratio table was developed uder cotract for the purpose of productio testig ad quality cotrol of motio sesitive sesors. The performace goal of the desig was to achieve a peak agular acceleratio of 5-6, /sec² ad to sythesize siusoidal stimulatio of the UUT up to a frequecy of 1KHz. Figure-1 is a mechaical assembly of the table ad idetifies the pricipal compoets. The motor is a 3 phase brushless desig which is mouted i a housig that is forced air cooled. The bearig system cosists of high precisio agular cotact ball bearigs that have bee customized for low frictio ad vibratory motio. A multi-pole positio sesor is mouted at the bottom ed of the shaft ad is used to close a cagig loop aroud the ier acceleratio loop ad supports positio cotrol i the.1 arc-mi rage. The table top is mouted to a relatively large diameter hollow shaft which esures a shaft resoace greater tha 3.5 KHz with the omial specified load of.1 Kgm². Fially, the acceleratio sesor is mouted i the ceter of the shaft ear the table top. This is also the ideal locatio for a commercial accelerometer as required i a traditioal vibratio table cofiguratio. 2. Commercial Accelerometers Let us first cosider the case of a rotary vibratio table that is costructed usig a commercial agular accelerometer for closig a high fidelity acceleratio loop. Agular accelerometers are desiged ad packaged i various styles ad are egieered to satisfy a variety of sesig applicatios. The preferred cofiguratio for a vibratio table sesor is a sigle housed uit with itegrated electroics ad suitable performace specificatios. The sesor package must be small ad closely coupled to the table top ad UUT. The axis of the sesor should be mouted cocetric to the table axis to miimize off axis sesitivity; therefore, the ideal locatio for the agular sesor is o the uderside of the table top i the hollow of the drivig shaft. A classical alterative sesig techique uses two liear accelerometers tagetially mouted at the outer limits of the table top. This approach ivolves meticulous electrical ad mechaical aligmets to esure that the two uits work as oe. The liear accelerometers must be very high quality with miimal sesitivity to the large radial acceleratios experieced durig peak rate operatio. The two uits must be idetically scaled, compesated ad aliged to avoid sesitivity to liear vibratios as i the case of multi-mode vibratio systems. A vibratio table that uses a commercial sesor for feedback caot perform ay better tha the measuremet sesor. The dyamic respose of the accelerometer is very critical sice it represets the differece betwee the measured ad the actual table acceleratio. The specificatios for cadidate sesors geerally reveal a fudametal tradeoff betwee amplitude respose (flatess) ad phase respose (phase shift) for uits of equivalet badwidth. This is best uderstood by

4 recogizig that a iertial accelerometer ca be reduced to a model cosistig of a mass or iertia ad a suspesio system or support elemet. The measure of acceleratio is proportioal to the torque or force actig o a referece body (α = T/ J ) as it is costraied to follow the motio of the base. Two thigs become obvious; all accelerometers (ope or closed loop) operate with fiite respose ad they exhibit mechaical resoace. A represetative performace specificatio established for the purpose of this paper is based o specificatios for various commercial agular accelerometers ad o product experiece. I a practical accelerometer desig, the dampig behavior critically effects the respose of the of the sesor i the frequecy rage of iterest. The required respose is based o the specified closed loop badwidth of ω = 1Hz for the vibratio table ad o the agular accelerometer havig a atural = 2 Hz. I the aalysis that follows, the accelerometer is modeled as a resoat mode at ω secod order dyamic system. The text book trasfer fuctio ; 1 G( jω) = 2 ω + j ζ ω ω ω is solved for the amplitude ad phase respose; Frequecy Respose Amplitude Ripple Phase KHz Udamped Natural Frequecy ; DC to 1 KHz ± 3 db ± 5 Deg. 2 KHz Mi. 2 ω Amplitude = LOG ζ ω 2 ω ω 2 2 ; Phase = 2ζ ω ω ta 1 2 ω 1 ω 2. The chart below plots the ormalized amplitude ad phase resposes as a fuctio of dampig ratio for the case where ω / ω = 5.. Note that at the poit where the amplitude respose is a miimum (ζ = 7. ) the phase shift is very large ad uacceptable. The poit where both the phase shift is below the 5 performace specificatio ad the amplitude is less tha 3 db, occurs at ζ = 5.. This suggests that a uder damped accelerometer respose with 2 db of peakig is eeded to meet the feed back requiremets over operatig frequecy rage. This peaked respose must be compesated i the cotrol system to esure closed loop stability.

5 Figure 2 3. Detailed descriptio of agular sesor as a accelerometer. 7 N o rm a liz e d 2 ND Order Respose for W/W=.5 (1KHz) Phase (Deg.) / 1X Magitude (db) A m p litu d e Pha s e The strai gage sesor was developed to fuctio as a idepedet acceleratio sesor ad be used i covetioal table desigs; Figure 3. I this cofiguratio the sesor offers limited advatage over covetioal accelerometers because it has similar dyamic characteristics. It is described here because it embodies the pricipals of operatio of the torque sesor. Dam pig Ratio (zeta) The sesor uit is costructed of 661-T6 alumium ad is machied etirely from oe cylidrical piece. The upper portio is the referece load iertia; 4 4 (( ) ( ) ) Jl =. 41 Dl / 2 dl / 2 Thk * Desity (i-lb-sec²); where: Dl = outer diameter of iertial load dl= ier diameter of the iertial load Thk = thickess of iertial load ad is coupled to the lower moutig base by a tubular sprig. This sprig is machied ito the sesig gap that separates the two portios of the sesor. The sprig costat of the tubular sprig is computed: 4 4 ( ) Ka =. 982G D d / l (i-lb/radia); where: G = modulus of elasticity; D = outer diameter of tubular sprig; d = ier diameter of tubular sprig; l = legth of tubular sprig. I the role of a accelerometer, the torque required to accelerate the iertial load is trasferred from the base through the sprig where the shear strai is cocetrated. The shear strai (γ ) that results o the iside surface of the sesig gap is extremely uiform ad liear: DT γ = 593. GD m 2 2 ( d ) (PSI);

6 where: Tm = torque of acceleratio. Figure 3 shows a istrumetatio grade strai gage boded to the torsioal sprig. The gage is comprised of two elemets o the same substrate which are orieted at ± 45 agles relative to the axis of sesitivity. A secod gage is diametrically mouted to complete a full strai gage differetial bridge. The aligmet of the strai gages is importat for cacelig liear or traslatioal couplig modes that are ot alog the sesitive axis. A high quality differetial amplifier is required to scale the sigal to a useable level. The output voltage of the strai gage amplifier is Vo=γ *GF*G Amplifier where: GF = the gage factor for the strai gage. Table 1 summarizes computed values of the equatios above based o various geometrical parameters, operatig coditios ad materials. The amplifier gai computed for the secod case is approximately 21, ad results i a sesitivity of 2 rad/sec²/volt. Accelerometer Aalysis Table 1 Material Al AL SS SS SS SS Desity Peak Accel. Ap ( /s^2) Twist Shaft D (i) d (i) l (i) Iertial Load Dl (i) dl (i) Thk (i) G (PSI) Iertia of Load Jl= Sprig Costat k= Resoat Frequecy w= Peak torque Tm= Shear Stress tau= Shear gama= 1.2E E-5 2.3E E E E-5 Double Bridge Voltage V1= Amplifier Gai (±5Vo) Gai= Detailed Descriptio of the agular sesor as a torque sesor. From the previous discussios it should be clear that the agular sesor is fudametally a torque measurig device ad i fact is very sesitive. The aalysis of Table 1 idicates that the peak torque measured for full scale output is oly.96 i-lb. The sprig costat of the torque sesor is TM 335, i-lb/rad ad the agular wid-up is computed as θ = K S 2.9 µ-radias. From this perspective, the sesor is capable of measurig very small positio displacemets. I cotrast, the stiffess of the output shaft has bee measured to be approximately 15x1 6 ftlb/rad ad at peak acceleratio trasfers 35 ft-lb of torque. The wid-up of this shaft is approximately 2.3 µ-radias which is spread uiformly across a large portio of the shaft legth. Referrig back to Figure 1 of the vibratio table, the agular sesor is securely mouted about half way dow the table shaft ad exteds early to the table top where it is coected with a viscous

7 couplig. The couplig is very stiff above 1 Hz, therefore the sesor appears to be very rigidly attached at both eds ad spas about 2.5 iches of the shaft. The uiform twist of the shaft over this spa is cocetrated i the gap of the torque sesor. The ratio of 2.5 ich spa to.15 ich sesig gap results i a mechaical gai of 16 to 1 or i other words it would require 16 times more electroic gai if the vibratio table torque were directly istrumeted with strai gages o the shaft. The viscous couplig o oe ed of the shaft provides a floatig mechaical coectio which reduces the precisio requiremets o the moutig iterfaces. It also prevets steady state wid-up of the sesor that could result i moutig ad temperature sesitive offsets i the measuremet. Aalysis 1. Plat Dyamic Model Figure 4 shows a lumped model of the plat which icludes the dyamics of the shaft resoace ad a dyamic model of a covetioal accelerometer. The iput to the plat model is the torque that is produced by the motor. Four outputs are plotted of which two are idetified as the scaled strai gage output ad the other is the commercial accelerometer output. The other two outputs are the acceleratio of the motor side ad table top side of the rotatig assembly. **** MODEL PARAMETERS **** Jm=.49; % i-lb-sec^2 Iertia ***** Kb1=25; MODEL % DEFINITION i-lb/rad/sec /Bearig ***** Dampig coeficiet % Ks=136;% ' A ' Matrix defiitio i-lb/rad Shaft Stiffess (lumped) a1=[-kb1/jm Jl=.72; %-Ks/Jm i-lb-sec^2 Kb1/Jm Table Ks/Jm Top Load Iertia ]; a2=[ Ja=.8; 1 % i-lb-sec^2 Accelerometer Equivalet Iertia ]; a3=[kb1/jl Ka=12; % Ks/Jl i-lb/rad -Kb1/Jl Stiffess -Ks/Jl of Accelerometer sesig ]; member a4=[ Kb2=1; % i-lb/rad/sec 1 Accelerometer Electroic Dampig ]; coeficiet a5=[ Kg=Ks/Jl; % Costat that coverts Ka/Ja accelerometer -Kb2/Ja -Ka/Ja]; strai to ormalized uits a6=[ 1 ]; am=[a1;a2;a3;a4;a5;a6]; % ' B ' Matrix defiitio bm=[1/jm ]'; % ' C ' Matrix defiitio c1=[-kb1/jm -Ks/Jm Kb1/Jm Ks/Jm ];% Acceleratio c2=[kb1/jl Ks/Jl -Kb1/Jl -Ks/Jl ]; % Table Top Acceleratio c3=[ Ka/Ja -Kb2/Ja -Ka/Ja];% Commercial Accelerometer c4=[ Ks/Jl -Ks/Jl ]; % output cm=[c1;c2;c3;c4]; % ' D ' Matrix defiitio dm=[1/jm;;;];

8 Magitude Db 5 Modeled Acceleratio Plat Figure Phase Deg 2 Accel Frequecy - Hz Accel. The motor side iertia (Jm) of the vibratio table exhibits a ati-resoace at 2.1 KHz where it resoates with the table side iertia (Jl) through the shaft stiffess (Ks). At 3.3KHz Jl resoates with Ks ad produces the characteristic secod order peaked respose. The accelerometer output icludes both the dyamics of the table ad the accelerometer which adds a resoat peak at 2 KHz. The strai gage output tracks the table top almost perfectly except at the high frequecies due to the effects of the atural viscous dampig (Kb1). The accelerometer adds peakig ad phase lag to the table respose. 2. Cotrol Strategy The cotrol architecture show i Figure 6 represets the closed loop servo i simplified form. The ier loop cosists of the domiat acceleratio loop which eeds to be appropriately compesated so that the performace goals are achieved. The outer positio loop maitais the ceter positio of the vibratory motio. This loop is compesated via Gp(S) ad has a very low badwidth to keep from disturbig the acceleratio respose at low frequecies. The outer positio loop is ot addressed. The plat trasfer fuctios provide the key iformatio to effectively compesate the closed acceleratio loops. Closig acceleratio loops aroud the commercial accelerometer ad the strai gage feedback requires a variety of compesatio filters which are represeted by Ga(S) i Figure 6. A

9 secod order low pass filter is added to the plat dyamics to model the 2 KHz badwidth of the power amplifier. The followig table summarizes the compesatio filters that are used for the two cases. The compesated ope loop trasfer fuctios are plotted i Figure 7 ad Figure 8: Compesatio Filter Commercial Accelerometer Sesor Power Amp (2 real poles) Wpa=2 Hz Wpa=2 Hz Shaft Resoace (Notch Filter) W1=3.3 KHz; ζ =.2 W1=3.3 KHz; ζ =.2 Accelerometer Resoace (Notch Filter) W1=2. KHz; ζ =.4 (oe) Itegrator With Lead Break Wl=6 Hz; Gai=142 db Wl=1 Hz; Gai=143 db Magitude Db -12 Compesated Acceleratio Loop Figure 7 (Commercial Accel) Magitude Db -12 Compesated Acceleratio Loop Figure 8 ( ) Accel. -24 Phase Deg Accel -5 Frequecy - Hz Phase Deg Accel -4-5 Accel. Frequecy - Hz The compesatio filters have bee adjusted so that the phase margi at crossover is greater tha 5 to esure that the closed loop peakig be kept to a miimum. The crossover frequecy for the accelerometer feedback loop is cosiderably less tha the strai gage loop as expected. The closed loop trasfer fuctios are plotted i Figure 9 ad Figure 1. The closed loop performace is summarized as follows: Performace Commercial Accelerometer Sesor Badwidth (9 phase) 6 Hz 15 Hz Badwidth (-3 db) 11 Hz 17 Hz Ripple ( to 1Hz) +, -3 db +, -.5 Sesor error (@ 1KKz) -2 db

10 Magitude Db Closed Acceleratio Loop Figure 9 (Modeled Accel) Magitude Closed Acceleratio Loop Figure 1 ( ) Accel -8 Phase Deg Table Table -4 Accel -5 Frequecy - Hz Phase Deg Frequecy - Hz Table Coclusios Vibratio table performace ca be sigificatly improved by elimiatig excess phase i each elemet of the servo system. The measuremet sesor is oly oe of may places where dyamic errors ca reduce performace, however, the sesor is the last word i reportig the true motio of the table top test surface. If the measuremet sesor is i error the o amout of closed loop badwidth will guaratee the fidelity of vibratory motio. The use of a strai gage sesor to measure the acceleratig torque actig o the free body load of a axis offers a alterative method to measure ad cotrol the acceleratio of a vibratio table. This techique provides a extremely liear, stable ad accurate measuremet with egligible added dyamics. Refereces 1. T. Kailath, Liear Systems,Pretice-Hall, Ic., Eglewood Cliffs, N.J., J. Schwartzebach ad K.F. Gill, Halsted Press, Joh Wiley & Sos, Ic., New York, C. Moler, J. Little & S. Bagert, PC-Matlab, Math Works, Ic. Sherbor, Ma, 1987