[American Institute of Aeronautics and Astronautics Control and Flight Dynamics Conference - Pasadena,CA,U.S.A. (12 August 1968 - 14 August 1968)] Control and Flight Dynamics Conference - Flotation technique for laboratory calibration of low-level accelerometers

v NO. 68-874 FLOTATION TECHNIQUE FOR LABORATORY CALIBRATION by BARRY K. LIKENESS and ROBERT H. CANNON, JR. Stanford University Stanford, California OF LOW-LEVEL ACCELEROMETERS W A I M Paper NO. 68-074 A lAA Guidance, Control, and Flighl Dynamics Conference v PASADENA, CALIFORNIA/AUGUST 12- 14, 1968 First publication riphls reserved by American lnrlilvte of A e r o n ~ ~ t i c ~ and A i l r o n w t i a . 1290 Avenue of the Americas. N e w 'fork. N. Y. 10019. Abstracts may be published withoul permittion if credit i s .iren lo aulhor and to AIAA. (Price: AlAA Member 11.00. Nonmember 11.501 FIOTATION TECHNIQUE FOR LABORATORY CALIBRATION OF LOW-LEVEL ACCELEROMETERS* B a r r y K . Likeness Research A s s i s t a n t Robert H . Cannon, Jr Prof ESSor Department of Aeronau t i c s and A s t m n o u t i c s S tan fo rd U n i v e r s i t y S t a n f o r d , C a l i f o r n i a A b s t r a c t A new l a b o r a t o r y t echn ique f o r e v a l u a t i n g low-level a c c e l e r o m e t e r s has been developed. The a c c e l e r o m e t e r proof mass is immersed in a dense gas which p a r t i a l l y s u p p ~ r t s i t s weight : t h e in s t rumen t i s then r e q u i r e d t o supp ly s u p p o r t f o r c e s comparable t o t h o s e expected in o r b i t . A t h r e e - a x i s e l e c t r i c a l l y suppor t ed acce le romete r (ESA) wi th a s p h e r i c a l hol low proof mass i s i d e a l l y s u i t e d f o r t h i s t echn ique . Choosing s u l i u r h e m f l u o r i d ? , a heavy gas , f o r t h e f l o t a t i o n f l u i d pe rmi t s a wide range of d e n s i t y levels Which can be servo- c o n t r o l l e d , and minimizes t h e e f f e c t of t h e f l u i d on t h e e l e c t r i c a l C h a r a c t e r i s t i c s of t h e in s t r i l - ment. A mathematical model Of t h e dynamics of t h e proof m a s s i n t h e p re sence of D f l o t a t i o n i l s L i d i s g i v e n , and i t s e f f e c t on c a l i b r a t i o n of t h e in s t rumen t is d i s c u s s e d . The pr-csence of t h e f l u i d has a n e g l i g i b l e e f f e c t on t h e c s l i b r o t i ~ n accuracy bu t does p reven t t h e s i m u l a t i o n of t r u e zero-g l i m i t - c y c l e behav io r . T h i s t echn ique c a n he u t i l i z e d w i t h s t a n d a r d d i v i d i n g head t echn iques . The accu racy of t h e s i m u l a t i o n depends upon t h e S t a b i l i t y of t h e f l o t a t i o n f l u i d d e n s i t y and t h e p r e c i s i o n w i t h which t h e in s t rumen t can be t i l t e d . Experimental r e s u l t s a r e g iven showing a d e n s i t y S t a b i l i t y of W i t h a p r e c i s i o n of i i l t changes of 10-6, i n p u t f o r c e s can be gene ra t ed Which are p r e c i s e to 10-13 g. For c a l i b r s t i o n purposes t h e in s t rumen t can be c o n v e n t i o n a l l y modeled. C o n t r o l l e d v a r i a t i o n of t h e f l o t a t i o n f l u i d d e n s i t y , a s well a s t h e t i l t a n g l e of t h e i n s t r u - ment, pe rmi t s Simultaneous d e t e r m i n a t i o n of s c a l e f a c t o r , b i a s e s , and c ross -coup l ing , w i thou t know- l e d g e of t h e a b s o l u t e vslue Of th? g a s d e n s i t y o r t h e o r i e n t a t i o n of t h e acce le romete r axes w i t h r e s p e c t t o t h e l a b o r a t o r y . I . INTRODUCTION The major a p p l i c a t i o n of acce le romete r s t o space programs today is moni to r ing of t h e h igh t h r u s t l e v e l s achieved during iaurrch ?row the ear th . ~Tiie a c c e l e r a t i o n levels range t y p i c a l l y from one g t o 15 g ; t h e s e acce le romete r s have been t e s t -p roven , b o t h i n t h e l a b o r a t o r y and o p e r a t i o n a l l y . Recent t r e n d s i n ae rospace r e s e a r c h have i n d i c a t e d a f u t u r e requirement f o r accelerometers t h a t w i l l measure a c c e l e r a t i o n levels t h a t may r ange from 10-2 g t o 10-12 g or even lower. Examples of v a r i o u s s p a c e mis s ions r e q u i r i n g measurements a t t h e s e levels a r e : i) Ontimum i n t e m l a n e t a r v t r a i e c t o r i e s w i l l r e q u i r e t h r u s t a c c e l e r a t i o n s . ranging from to 10-5 g . i i ) Exo t i c ( e . g . , e l e c t r i c , nuclear) propu l s ion + Work suppor t ed i n p a r t by A i r Force Con t rac t F33615-67-C-1245, u n i t s f o r deep space mis s ions will develop t h r u s t a c c e l e r a t i o n s from e q u i v a l e n t a c c e l e r a t i o n l e v e l s from 111-6 g g t o lo-' g . i i i ) Grav i ty -g rad ien t measurements will have LO 10-8 g. i v ) T r a j e c t o r y p e r t u r b a t i o n s due t o e x t e r n a l d i s t u r b a n c e s Such a s aerodynamic nnd solar p r e s s u r e , magnet ic and electrical i n t e r - a c t i o n s w i t h ambient f i e l d s , rnicrometeor- i t e s , e t c . , and v e h i c l e dynamics will produce a c c e l e r a t i o n s r ang ing from lo-' g 10 10-12 g. An o r b i t a l o r e q u i v a l e n t l a b o r a t o r y t e s t nust v e r i f y in s t rumen t performance over a major p o r t i o n of t h i s range of magnitudes. The b a s i c l i m i t a t i o n t o l a b o r a t o r y c a l i b r a t i o n t echn iques of low-level acce le romete r s i s t h e presence of t h o e a r t h ' s g r a v i t a t i o n a l f i e l d . ( I ) Th i s f i e l d i n t r o d u c e s B one-g f o r c e o r e q u i v a l e n t a c c e l e r a t i o n i n t h e d i r e c t i o n of t h e l a b o r a t o r y local V e r t i c a l , which must be compensated f o r by r e b a l a n c e f o r c e s gene ra t ed by t h e acce le romete r . Th i s r e b a l a n c e or Support f o r c e will i n t r o d u c e errors and u n c e r t a i n t i e s due t o t h e n o w u n i f o r m i t y t h e i n s t r u m e n t ' s S e n s i t i v e axes . Present-day s t a t e - o f - t h e - a r t f a b r i c a t i o n t echn iques con reduce t h e s e errors t o a level t y p i c a l l y from g t o l O + g, and t o s t procedures have been developed t h a t can de te rmine t h e i r va lue t o a p r e c i s i o n approaching g . ( 2 ) Of t h e s u p p o r t f o r c e , and t h e "on-orthogonal i ty of \.' These performance levels f a l l s h o r t of t h e d e s i r e d 10'' g t o lo-' g levels t h a t w i l l soon be r e q u i r e d f o r t h i s c l a s s of in s t rumen t . I n a d d i t i o n , d u r i n g t h e s e c a l i b r a t i o n p rocedures , t h e in s t rumen t i s a c t u a l l y o p e r a t i n g a t a one-g level. T h i s is many o r d e r s of magnitude l a r g e r t han will bc encountered d u r i n g a space mission: t hus t h e in s t rumen t would be r e s c a l e d b e f o r e launch t o o p e r a t e a t t h c lower levels expec ted . T h i s r e s c a l i n g procedure may i n v a l i d a t e t h e c a l i b r a t i o n r e s u l t s o b t a i n e d w i t h t h e in s t rumen t o p e r a t i n g a t a one-g l e v e l . For t h e s e reasons i t is d e s i r a b l e t o compensate f o r t h e e a r t h ' s one-g f o r c e wi th a r e b a l a n c e force t h a t i s independent of t h e accelerometer. The ins t rumen t cou ld then be c a l i b r a t e d w h i l e o p e r a t i n g a t t h e Same force levels t h a t Would be ehcountercd d u r i n g a space miss ion . T h i s has been achicvcd by immersing t h e acce le romete r proof mass in a s t a t i o n a r y , dense gas . A s Archimedes' P r i n c i p l e s t a t e s , t h e p re sence of t h e gas produces a n e t f o r c e on t h e proof mass which is a n t i p a r a l l e l t o t h e e a r t h ' s g r a v i t a t i o n a l f o r c e and i s equal i n magnitude t o t h e weight Of t h e gas d i s p l a & ? d . T h e level Of performance t h a t c a n be ach ieved is determined by t h e p r o p e r t i e s of t h e gas o r W 1 f l o t a t i o n f l u i d , r a t h e r t han l i m i t e d by u n c e r t a i n - t i c s i n t h e accc lo romete r i t s e l f . 11. PROPERTIES OF BUOYANCY I f a body is immersed i n a s t a t i o n a r y f l u i d , t h e n e t f o r c e a c t i n g on i t w i l l equal t h e n e g a t i v e of t h e weight of t h e f l u i d i t d i s p l a c e s . The buoyancy force i? can t h e r e f o r e be w r i t t e n as PROPERTY For a proof mass t ha t+ i s m a 1 1 wi th r e s p e c t t o t h e r a d i u s of t h e e a r t h , g(v) can be cons ide red as c o n s t a n t over t ho volume of t h e f l u i d d i s p l a c e d . Rep lac ing d v ) by an e q u i v a l e n t mean d e n s i t y pf f o r t h e f l u i d d i s p l a c e d , t LIQUIDS GASES where Vm 16 t h e volume of t h e proof mass. The m r t h ' s one-g g r a v i t a t i o n a l f i e l d produces a f o r c e on t h e proof mass given by Heat Conduc t iv i ty i w h e r e w i s known as t h e "weight" of t h e proof mass. Again c o n s i d e r i n g 3 t o be c o n s t a n t Over t h o volume or t h ~ proof mass, and r e p l a c i n g p ("1 by a n e q u i v a l e n t mean d e n s i t y F f o r themproof mass, ii 3 w = 6 iii, Crn dv = FmVmg The n e t f o r c e a c t i n g on t h e proof mass due to t h e g r a v i t a t i o n a l f i e l d and t h e f l o t a t i o n f l u i d i s then 3 = ? + C poor n e g l i g i b l e where B = (1 - Ff/Fm) is d e f i n e d 85 t ho f l o t a t i o n f a c t o r . When of and om are c o n s t a n t over t h e volume of t h e proof mass, t h e n e t f o r c e 3 w i l l a c t a t t h e Cen te r Of mass of t h e proof mass. S l i g h t v a r i a t i o n s i n t h e d e y i t y may s h i f t t h e c e n t e r of t h e buoyancy v e c t o r b from t h e c e n t e r of t h e g r a v i t a t i o n a l force v e c t o r W. This i n t r o d u c e s a t o rque on-the pgoof mass t h a t w i l l be p r e s e n t whenever b and w are no t c o l i n e a r - . S ince damping will be p r e s e n t due t o t h e v i s c o s i t y of t h e f l o t a - t i o n f l u i d , t h o proof mass w i l l s eek an e q u i l i b r i u m p o s i t i o n w i t h b and w c o l i n e a r . + - 3 Some of t h e p r o p e r t i e s of t h e buoyancy f o r c e a r e immediately advantageous f o r t e s t i n g en acce le ro - meter i n a l a b o r a t o r y . These p r o p e r t i e s i nc lude : Thc buoyancy fo rce f i e l d i s a n t i p a r a l l e l t o t h e l o c a l g r a v i t a t i o n a l f i e l d . The i n s t r u - ment s u p p o r t f i e l d can t h u s be s u b s t a n t i a l l y r e p l a c e d by a f o r c e t h a t remains a n t i p a r a l l e l +The g r a d i e n t i n 3(v) a t t h e S u r f a c e of t h e e a r t h i s 3.1 X lo-' g/cm i n t h e v e r t i c a l d i r e c t i o n and 1 .6 X lo-' g/cm i n t h e h o r i z o n t a l d i r e c t i o n . t o t h e e a p t h ' s one-g f i e l d independent of f l u c t u a t i o n S i n t h e p r o p e r t i e s o f t h e i n s t r u - ment. can b e reduced by t h e f l o t a t i o n f a c t o r , B , t h u s a t t e n u a t i n g t h e a b s o l u t e rnagnltude of t h e errors due t o t h e Don-uniformity of t h e Support f i e l d s and t h e non-o r thogona l i ty of t h e a c c e l e r o m e t e r exe5. The n e c e s s a r y in s t rumen t s u p p o r t f i e l d ; 0 By c o n t r o l l i n g t h e d e n s i t y of t h e f l o t a t i o n f l u i d , t h e f l o t a t i o n f a c t o r , B, can be made p o s i t i v e or n e g a t i v e . T h i s r e v e r s e s t h e e f f e c t i v e g r a v i t a t i o n a l f i e l d and i s equiva- l e n t t o r o t a t i o n s of t h e in s t rumen t through B i s a n a t t e n u a t i o n f a c t o r . All g r a v i t a t i o n a l or i n e r t i a l i n p u t s t o t h e a c c e l e r o m e t c r a r e a t t e n u a t e d by 6 . I n p a r t i c u l a r , t h e d e s i r e d i n p u t s must be i n c ~ l e a s e d by B-', r a i s i n g them above t h e l e v e l of t h e e a r t h ' s s e i s m i c n o i s e . F o r an e f f e c t i v e i n p u t o f IO@ g w i t h 3 B of e x a c t l y 1800. t h e a c t u a l i n p u t must bo where F is t h e d e s i r e d i n p u t to t h e proof mass and I is t h e r e q u i r e d i n p u t to t h e s y s ten]. 0 - 1.0 0.6 - 5 . 0 I a t h igh p D e n s i t i e s C o m p r e s s i b i l i t y 1 ntm-' 1 P = kP Temperature 1 i? 0.0005 - 0.002 0.003 - 0.01 I S e n s i t i v i t y i g 1 "c-1 I ac-1 V l s c o s l t y moderate I t o h igh D i e l e c t r i c Cons tan t (vacuum = 1) 1 2 - 100 1 1.0 - 1.1 TABLE 1 PROPERTIES OF GASES AND LIQUIDS Both l i q u i d s and gases ore p o s s i b l e c a n d i d a t e s for t h e f l o t a t i o n f l u i d . A summary of t h e i r prop- e r t i e s Which have a r e l a t i o n s h i p t o t h e f l o t a t i o n t echn ique i s g iven i n Tab le 1. A f t e r reviewing the5e p r o p e r t i e s and c o n s i d e r i n g o t h e r f a c t o r s Such a s Cost, a v a i l a b i l i t y , s a f e t y , e t c . , S u l f u r h e x a f l u o r i d e (SF 1, a c o l o r l e s s , o d o r l e s s , heavy gas w i t h a h igh S i e l e c t r i c S t r e n g t h Was chosen a s t h e f l o t a t i o n f l u i d . The d e n s i t y of t h e gas is determined by i t s pressure and t empera tu re ; F i g u r e 1 shows t h e l e v e l s of d e n s i t y t h a t Can be ach ieved u s i n g p,rressures up t o 1000 p s i a . 68 - 814 2 FIGURE 1 ISOTHERMAL PRESSURE/DENSITY CURVES FOR SF6 111. DENSITY MEASUREMEWT AND CONTROL OF THE FWTATION FACTOR B From Eq. (l), t h e f l o t a t i o n f a c t o r B i s dependent Only on t h e d e n s i t i e s of t h e f l o t a t i o n f l u i d and t h e proof mass. The proof mass volume, and t h e r e f o r e its d e n s i t y , w i l l b e S e n s i t i v e t o changes i n p r e s s u r e and t empera tu re , b u t t h i s e f f e c t w i l l be s e v e r a l o r d e r s of magnitude less t han t h e s e n s i t i v i t y of t h e f l o t a t i o n f l u i d d e n s i t y t o t h e s e changes. The d e n s i t y of t h e p ~ o o f mass can t h e r e f o r e be cons ide red c o n s t a n t and v a r i a t i o n s i n t h e f l o t a t i o n f a c t o r B w i l l b e determined by v a r i a t i o n s i n t h e d e n s i t y of t h e f l o t a t i o n f l u i d , 8 5 i n d i c a t e d by Eq. (2): * P f Pm A B = - - The g e n e r a t i n g of a c o n s t a n t f l o t a t i o n f a c t o r B i s t h u s r e so lved i n t o t h e problem of measuring and c o n t r o l l i n g a Constant d e n s i t y of t h e f l o t a t i o n f l u i d . ' When t h e f l o t a t i o n f l u i d is a gas and i s con ta ined i n a s e a l e d Con ta ine r ( t h e f l o t a t i o n chamber), v a r i a t i o n s i n t h e mean d e n s i t y of t h e f l o t a t i o n gas throughout t h e chamber w i l l o r i g i n a t e from V a r i a t i o n s i n t h e volume of t h e chamber and l eakage of t h e f l o t a t i o n gas from t h e chamber. By proper d e s i g n , t h e pressure S e n s i t i v i t y of t h e .chamber 's volume, and t h e l eakage of t h e f l o t a d i o n gas can be made a r b i t r a r i l y s m a l l . Temperature c o n t r o l is r e q u i r e d , however, t o minimize t h e e f f e c t s of t empera tu re on t h e volume of t h e f l o t a - t i o n chamber, and t o keep t h e f l o t a t i o n gas from l i q u i f y i n g . S i n c e t h e f l o t a t i o n gas i s a very poor thermal conduc to r , and s i n c e t h e l o c a l d e n s i t y is S e n s i t i v e t o l o c a l t empera tu re changes, c a r e must he t aken t o keep l o c a l i z e d h e a t sources and t empera tu re v a r i a t i o n s t o a minimum. Densi ty Measurements I n o r d e r t o d e t e c t changes i n d e n s i t y a s m a 1 1 a s one p a r t i n one m i l l i o n , i t is q u i c k l y appa ren t t h a t conven t iona l methods of measuFing t h e d e n s i t y weighing a proof mass i n t h e f l o t a t i o n chamber. S i n c e its "weight" w i l l b e reduced by t h e buoyancy force a c t i n g on i t , Which is d i r e c t l y p r o p o r t i o n a l t o t h e d e n s i t y of t h e f l o t a t i o n gas , t h e d e n s i t y of t h e gas c a n b e determined by measuring t h e e f f e c t i v e weight of t h e d e n s i t y sensor proof mass must b e abandoned. The method chosen c o n s i s t s of U where w ' is t h e e f f e c t i v e weight ijm is t h e mean d e n s i t y w i s t h e weight i n a vacuum of t h e d e n s i t y sensor proof mass pf is t h e mean d e n s i t y of t h e f l o t a t i o n g a s so t h a t is, is known a s a c c u r a t e l y a s measurements of pm, w ' and w can be made. Moreover, so t h a t r e l a t i v e changes i n pf can be measured a 5 a c c u r a t e l y a s t h e measurements of Aw' and w will permi t . When t h e d e n s i t i e s Of t h e f l o t a t i o n gas and t h e proof mas5 are very n e a r l y e q u a l , c h o i c e of t h e ba l ance S e n s i t i v i t y Aw' and t h e weight w of t h e d e n s i t y sensor proof mas5 can make t h i s d e n s i t y senso r a r b i t r a r i l y s e n s i t i v e . Densi ty Servo-Control ii Since gas i s many o r d e r s of magnitude more compress ib l e than s o l i d s or l i q u i d s , t h e mean d e n s i t y of t h e gas i n t h e f l o t a t i o n chamber can be e f f e c t i v e l y Con t ro l l ed by va ry ing t h e volume of t h e chamber. T h i s is achieved by h y d r a u l i c a l l y c o n t r o l l i n g t h e expansion of a bel lows Which i n t e r f a c e s wi th t h e chamber (see F i g u r e 2). Any change i n t h e e f f e c t i v e weight of t h e d e n s i t y sensor proof mass can be used t o a c t i v a t e t h e h y d r a u l i c mechanism t o vary t h e volume of t h e chamber u n t i l t h e e f f e c t i v e weight has r e t u r n e d to its O r i g i n a l v a l u e . T h i s i s known a s B servo- c o n t r o l l e d d e n s i t y loop and 1s ana lyzed by t h e s t a n d a r d t echn iques of se rvo-con t ro l t heo ry . A c t i v a t i o n of t h e Servo-control loop w i l l keep t h e d e n s i t y of t h e gas c o n s t a n t a t t h e l o c a t i o n of t h e d e n s i t y sensop. A5 l ong a s g r a d i e n t s remain c o n s t a n t and convec t ive e f f e c t s a r e n e g l i g i b l e , t h e servo w i l l keep t h e d e n s i t y c o n s t a n t t h ro lgh - o u t t h e f l o t a t i o n chamber w i t h i n t h e limits of t h e se rvo-con t ro l mechanism. Flotation Chamber (Figure 2) The flotation chamber is essentially an aluminum cylinder with removable end plates. The chamber was designed to operate at pressures up to 750 psia and temperatures up to 60' C, thus permitting fluid densities up to 1.0 when using sulfur hexafluoride (see Figure 1). The internal volume of the chamber is approximately 600 cubic inches (9000 c c ) . One end plate contains the flotation fluid inlet/outlet port, hermetically-sealed high pressure electrical feedthroughs, and provision for mounting a bellow8 element that is subsequently utilized in the servo-controlled-density mechanization, together with its associated hydraulic inlet/outlet port. The body and the two end-plates are constructed of 6061 T-6 aluminum. An interchangeable third end plate, Constructed of carbon steel, has an observation window mounted in it. All mating surfaces are sealed with O-rings. Density Sensors The density sensors are commercial microbalances that are employed to read the effective weight of proof masses directly. The balances operate on the principle of direct Current torque rebalance, and Produce dc output voltages that are proportional to the effective weight Of the proof masses. Recording the output Voltage thus provides continuous information on the density of the flotation fluid at the position of the proof mass, assuming that the density of the proof mass remains constant. Operatine two independent density 1 2 3 Flotation Chamber (Pressure Vessel) Density Sensor Density Sensor Proof Mass Bellows Hydraulic Fluid Heater Temperature Control Circuitry Thermal Insulation Observation Window FIGURE 2 FWTATION CHAMBER IV. THE FWTATION FACILITY . sensors simultaneously permits an investigation of the spatial density gradients that may be present Selection of either density Sensor output to drive the volume servo-controller. To demonStrate the Of the technique, an experimental program was undertaken, in the flotation culminating in an actual prototype model of a flotation test facility. of the program was to create and measure the An option permits The primary objective ~~ performance Of a high quality density environment. Figure 2 illustrates the flotation chamber. The functional relationship of the required Support equipment is Shown in Figllre 3; a brief descrip- tion of some of the principal components follows. The ultimate sensitivity of the microbalance is + 0.1 X gram. Using a 5 gram proof mass, the sensitivity of the microbalance 8 5 a density Sensor is - I FIGURE 3 SDW-G ENVIRONMENT FUNCTIONAL DIAGRMI 4 68 - 8'14 or two p a r t s pe r 100 m i l l i o n ; i t s f r equency r ange is from d c t o 10 r a d i a n s pe r second. F l o t a t i o n Chamber Volume C o n t r o l l e r The o u t p u t of t h e d e n s i t y sensor, a dc v o l t a g e t h a t i s p r o p o r t i o n a l t o t h e e f f e c t i v e weight of t h e proof mass, is used t o a c t i v a t e t h e hydTBUliCBlly-aCtUBted Servo-mechanism t h a t c o n t r o l s t h e expansion of a be l lows mounted i n t h e f l o t a t i o n chnmber. T h i s vollies t h e e f f e c t i v e volume of t h e chamber; S i n c e t h e mass of f l o t a - t i o n gas con ta ined i n t h e f l o t a t i o n chamber i s . ' Cons tan t , t h e expansion of t h e bel lows v a r i e s t h e d e n s i t y of t h e f l o t a t i o n gos. The h y d r a u l i c p i s t o n which d r i v e s t h e f l u i d i n t o t h e be l lows chamber i s d r i v e n by B s t e p p e r motor Whose r a t e i s p r o p o r t i o n a l t o t h e o u t p u t of tlic d e n s i t y sensor. One s t e p i s e q u i v a l e n t t o a change i n volume Of 25 x Cubic inches : t h i s C B U S ~ S a change i n t h e d e n s i t y of t h e f l o t a t i o n gas .o f Of f o u r p a r t s pe r 100 m i l l i o n . The maximum r a t e of t h e s t e p p e r motor i s 250 s t e p s p e r second or and i t s range is 5000 s t e p s or Temperature C o n t r o l l e r s The i l o t a t i o n chaiober t empera tu re c o n t r o l l e r m a i n t a i n s t h e t empera tu re of t h e f l o t a t i o n chamber and t h e f l o t a t i o n f l u i d a t a c o n s t a n t p r e s e l e c t e d t empera tu re between 30' C and 60' C. I t5 b a s i c f u n c t i o n s are t o 0 t o p reven t t h e f l o t a t i o n gas from t o r educe convec t ion c u r r e n t s i n t h e 0 to keep t h e volume of t h e chamber l i q u i f y i n g f l o t a t i o n gas S t r u c t u r e c o n s t a n t , t hus m a i n t a i n i n g a Constant mean d e n s i t y i n t h e f l o t a t i o n gas . The h e a t e r is a c o i l of copper w i r e , approximately 500 f e e t i n l e n g t h , wound c i r c u m f e r e n t i a l l y and non- induc t ive ly abou t t h e f l o t a t i o n chamber. The t empera tu re is sensed by a S o l i d S t a t e t h e r m i s t o r l o c a t e d on t h e e x t e r n a l s u r f a c e Of t h e f l o t a t i o n chamber, i n a d c b r i d g e network. The chamber i s well i n s u l a t e d w i t h s ty ro foam, r a d i a t i n g 3 w a t t s when o p e r a t e d a t 20.' C above room tempera tu re . The s h o r t term t empera tu re v a r i a t i o n s a r e less t han O.O0lo C a t t h e l o c a t i o n o f t h e t h e r m i s t o r , and less t h a n 0.0001' C i n t h e i n t e r i o r of t h e f l o t a t i o n chamber. V. EXPERIMEWAL RESULTS The d e n s i t y performance throughout t h e f l o t a t i o n chamber was i n v e s t i g a t e d by s imul t aneous ly r e c o r d i n g t h e o u t p u t of t w o independent d e n s i t y sensors mounted a t v a r i o u s l o c a t i o n s throughout t h e chamber. A t y p i c a l experimental run i s i l l u s t r a t e d i n F i g u r e 4. W I n i t i a l l y ( r eg ion A ) t h e Servo-Control i s i n a c t i v e , and t h e two independent d e n s i t y sensors r e c o r d t h e v a r i a t i o n s i n t h e f l o t a t i o n f o r c e s a c t i n g on t h e proof masses a t each l o c a t i o n . Due t o t h e mounting Conf igu re t ion of t h e two d e n s i t y sensors i n t h i s case, t h e o u t p u t s of t h e two sensors are of o p p o s i t e sense. Thus a slow d e c r e a s e i n t h e mean d e n s i t y of t h e f l o t a t i o n gas (an i n c r e a s e i n t h e e f f e c t i v e weight of each proof mass) i s i n d i c a t e d by an upward d r i f t of t h e ou tpu t of sensor #1 and B downward d r i f t i n t h e o u t p u t of sensor # Z . The p e r i o d i c f l u c t u a t i o n s (15, 30, and 60 Second p e r i o d s ) a r e seen t o be of t h e o r d e r of p l u s or minus 5 micrograms, Which co r re sponds 1.0 v a r i a t i o n s i n t h e f l o t a t i o n f a c t o r B of p l u s or minus one p a r t pe r m i l l i o n . I n r eg ion 8 , t h e o u t p u t of d e n s i t y sensor No. 1 i s used t o c o n t r o l t h e mean d e n s i t y of t h e f l o t a - t i o n gas . I t i s seen t h a t t h e long-term d r i f t h a s boen e l i m i n a t e d a t bo th l o c a t i o n s , bu t t h a t w h i l e shDrt pe r iod v a r i a t i o n s a r c reduced a t l o c a t i o n No. 1, they are i n c r e a s e d by about 40$ u t l o c a t i o n N o . 2 . The v a r i a t i o n s i n t h e f l o t a t i o n f o r c e a t l o c a t i o n No. 1 a r e reduced t o p l u s o r minus B few p a r t s i n t e n m i l l i o n . I n r eg ion C , t h e ro les of t h e two d e n s i t y sensors a r e r e v e r s e d , and t h e r e s u l t s a r e s i m i l a r t o t h o s e i n r e g i o n B . The r e s u l t s of t h i s phase of t h e experiment81 L, i n v e s t i g a t i o n i n d i c a t e s t h a t t h e s e r v o - c o n t r o l l e d f l o t a t i o n performance t h a t can be ach ieved a t a remote l o c a t i o n i n t h e chamber i s approxi- mately p l u s or minus one or two p a r t s p e r m i l l i o n , w h i l e t h e e f f e c t i v e f l o t a t i o n c o n t r o l a t t h e l o c a t i o n of t h e a c t i v e d e n s i t y sensor is a n o r d e r of magnitude b e t t e r , or p l u s or minus a few p a r t s pe r t e n m i l l i o n . While a few p a r t s pe r m i l l i o n 1s an a c c e p t a b l e performance level, t h e r e a r e a d d i t i o n a l t echn iques a v a i l a b l e for improved r e s u l t s . S i n c e t h e long pe r iod v a r i a t i o n s i n t h e f l o t s - t i o n f o r c e s t t h e remote l o c a t i o n a r e e l i m i n a t e d by t h e s e r v o - c o n t r o l l e r , t h e ave rage va lue of t h e v a r i a t i o n s i n t h e f l o t a t i o n f a c t o r B i s h e l d t o zero. Using a v a l u e f o r B t h a t i s ob ta ined by a v e r a g i n g , r a t h e r t h a n t h e i n s t a n t a n e o u s v a l u e w i l l improve t h e r e s u l t s of c a l c u l a t i o n s dependent on t h e v a l u e of B . The e f f e c t i v e f l o t a t i o n f o r c e i s comprised of t h e h y d r o s t a t i c f l u i d f o r c e which e x h i b i t s l ong pe r iod v a r i a t i o n s , and v i s c o u s d r a g forces c r e a t e d by convec t ion c u r r e n t s f lowing p a s t t h e su ' r facc of t h e d e n s i t y sensor proof mass, c a u s i n g s h o r t pe r iod v a r i a t i o n s i n t h e e f f e c t i v e weight of t h e proof mass. Surrounding t h e d e n s i t y sensor by a housing, o r o t h e r w i s e b a f f l i n g t h e f l u i d flow near t h e d e n s i t y sensor, w i l l r educe t h e s h o r t p e r i o d v a r i a t i o n s , l e a v i n g t h e l o n g e r pe r iod v a r i a t i o n s which a r e reduced throughout t h e f l o t a t i o n Chalnber by t h e volume S e r v o - c o n t r o l l e r . L' TO measure t h e expected improvement i n t h e FIGURE 4 FKWATION CHAMBER DENSITY PERFORhlANCE Density Sensor Proof Masses: 5 gm Effective Weight: Flotation Gas Temperature: 38' C Density: 0.202 gm/cc 2 mgm (B = 4 X l(r4) FIGURE 5 THREE-AXIS MEASUREMENTS OF FLOTATION FLUID FORCES Accelerometer Proof Mass: 5 gm Effective Weight: 0 w Flotation Gas Temperature: 38' C Density: 0.202 m/cc 6 68 - 874 vertical force environment due to the presence of a housing, and to measure the horizontal forces acting on the density sensor, a three-axis low level dc electrostatic accelerometer utilizing a spherical proof mass was constructed. The instru- ment has a preload of IO-' g, a sensitivity of lo-' g and a frequency response of dc to 0.1 rad/sec. With the remote density sensor controll- ing the flotation fluid density, the outputs of the instrument's three axes (one vertical, two horizontal) were simultaneously recorded. Typical results are shown in Figure 5. The variation in the vertical axis is seen to be plus or minus one part in 10'; in the horizontal axes the variation is approximately a few parts in 10'. variations are at frequencies much greater than the frequencies associated with fluid disturbances and are due to the limitation in the accelero- meter's sensitivity. An upper limit On the magni- tude of the fluid disturbances in the horizontal axes can be placed at an equivalent of IO-' 2. Use of more Sensitive accelerometers could revea l that the horizontal force levels a r e much smaller than this upper limit. V I . EFFECTS OF FUJTATION ON INSTRllMEm PERFORMANCE These In principle, many different types of accelero- meters are redesignable for orbital applications. However, the flotation technique for calibrating orbital accelerometers is restiicted to a ma11 number of instruments. The principal requirements are: i ) The proof mass must be susceptive to immersion in a flotation fluid. ii) The instrument must have nominally Stationary components. iii) Heat sources inherent in the operation of the instrument must be minimized. Of the instruments currently proposed for orbital applications, the type that presently appears most suitable for calibration with the flotation technique is the electrostatic accelero- meter. One particular instrument in this class is a three-axis accelerometer utilizing a spherical proof mass enclosed in a Spherical cavity. Conducting plates deposited on the wall of the cavity are located along three orthogonal axes and are used to capacitively sense the &ition of the ball and to electrostatically force the proof mass to remain centered in the cavity. Determination of the electrostatic force5 yields information on the external forces acting on the proof mass. Before the instrument can be calibrated with the flotation technique, an accurate model of the effect of the presence of the flotation fluid on the instrument's performance m u s t be obtained. The effects of the flotation fluid on the instru- ment can be placed in two categories, static and dynamic. Static Effects Using a pressurized gas at B n elevated tempera- ture will cause dimensional changes i n the instru- m e n t . These effects can be predicted from the measured va lues of the physical properties of the materials from which the instrument is made. In addition, the dielectric Constant of the flotation gas will be different from unity; this quantity can be accurately determined by measuring the change in the value Of the capacity of an air-core capacitor in the flotation fluid from its v a l u e in a vacuum. Dynamic Effects v Although the accelerometer's proof mass is nominally stationary in the center of the cavity, any time variation in the forces acting an the proof maas will result in Some residual motion of the proof mass relative to the cavity. This will produce additional forces acting on the Proof mass due to inertia and shear forces developed in the flotation fluid. Determination of these forces 15 a problem of fluid mechanics, a field in which analytical solutions a r e often rare . The most general equations of the.motion of a viscous, compressible fluid are the Navier-Stokes equations Which a r e nonlinear, time-varying, partial differential equatiom. These equations can be linearized by assuming i) the fluid is incompressible ii) velocities a r e small so that products Of velocities can be neglected. The boundary conditions required to determine Solutions to the resulting partial differential equations ere derived from the assumption that the fluid is stationary relative to any geometric boundary. Consider a viscous, incompressible fluid bounded by two concentric spherical surfaces, the inner one oscillating about the midpoint of the outer one. Since the fluid is considered incompressible, i t must be continually accelerated due to the relative motion of the t w o boundaries. The inertia of the fluid and the viscous dissipation will result in a force acting on the oscillating sphere. d This force has been derived by Stokes(3) for the case of Simple harmonic motion of the inner spheri- cal surface. The formula derived is where M is the 111855 of the fluid "displaced" by the inner sphere relative to the Outer sphere, restricted t o sinusoidal motion V is the velocity of the inner sphere a,b are the radii of the inner am! outer sphere u' = frequency in radians/second p = density Of the fluid IJ. = viscosity of the fluid i =,/-I, an operator 2 2 . 2 2 m (b-ii) q ( a , b ) = (rn a +3mot3)(m b -3mb+3)e 2 2 2 2 L ( G , i ) ) = [b(m b -3mb+3)-a(m B +3ma+3)]em(b-3) L/ If a Sinusoidal force F(t) is applied to the inner sphere, the resulting steady state ination w i l l then be determined from I where M 1s t h e mass of t h e s p h e r e X 1s t h e d i sp lacemen t of t h e s p h e r e When t h e inner s p h e r e i s almost n e u t r a l l y buoyant , Eq. (4) becomes M Z M. Since - = -, dV d2X " d t 2 f S t Assuming X = X e , F = Foest, wit'h s = io, Thus a t r a n s f e r f u n c t i o n r e l a t i n g Xo t o Fo can be w r i t t e n : The e f f e c t of t h e presence of a viscous f l u i d and a n a d d i t i o n a l c o n c e n t r i c s p h e r i c a l boundary i s a m o d i f i c a t i o n of t h e b a s i c - t r a n s f e r f u n c t i o n of t h e i n n e r s p h e r e . 1 MS 2 When (b-a) Figure 7 From Figure S ( a ) , FIGURE 7 BODE PIOT OF ESA PROOF MASS The effect of the modified dynamic model on the performance of the instrument can be determined from Figures 8 0 ) and 8(b). FIGURE 8 ( a ) PROOF &!ASS hl IN VACUUM FIGURE 8(b) PROOF MASS hl IN FLarATlON FLUID where F ( s ) is the input force X ( s ) is the proof mass position H ( s ) is the accelerometer control and compensation Facc(s ) is the accelerometer restoring force. From Figure 8(b), 2 - F ( s ) when l H ( s ) l >> IZOMs(s+l)l Thus the modified dynamic model of the proof mass will not affect the performance of the accelero- meter when the inputs are constant or at very l o w frequencies. However, a study must be made of the effect this model will have on the control and compensation loops designed into the accelerometer to ensure stable Operation in the presence of the flotation fluid. VII. CALIBIUTION TECHNIQUES The flotation chamber can readily be utilized in conjunction with standard dividing head procedures to provide a very l o w l e v e l calibration technique for accelerometers which a r e susceptible 10 i l o t n - tion. For example, if the Spherical proof mass immersed in the stationary flotation fluid, the net force acting on it will be its effective weight, which is equal to the product of the nominal weight mzand the flotation factor E . the Sensitive axes of the accelerometer are oriented a s Shown in Figllre 9 , (the y - a x i s is horizontal) the components of the a~celeromoter force Face will be of in three-axis electrostatic accelerometer is L’ I f --f e x '"i I( EFFECTIJI WEIGHT FIGURE 9 COMPONENTS OF ACCELEROMETER FORCES Normalizing the indicated force levels with respect to a one-g force acting on the proof mass, the horizontal input can be written $x+Fxe = [l%'i3^ e$+&e+Beee ]g (7) where B," and Be a r e errors in the estimated va lues - 8 and 0 from their actual v a l u e s B and e . The precision of f will be determined by tne magnitude of the last three terms in Eq. (7). The magnitudes 01 Be and e, due to the limitations of their respective servo-control loops are and error terms are then respectively. The magnitudes of the three &, B* . g, g. The relationship of these errors to the magnitude Of the input is conveniently illustrated by Figure 10. The principal error terms are found at the intersection of horizontal a,n$ vertical lines through the operating point B , e with the vertical e, line and the horizoqtzj: Be l i n e . magnitude of both the input term @ and the principal error terms are read from the corre- sponding diagonal constant force lines. ,For example, With an operating condition of B = 10-5, 6 = lo-*, the input ix 10-9 g, and the errors are Thus an input of g can be generated with a rms error of 1.4$. that a "zero" input can be produced in the hori- zontal axis with a precision of 1 O - l 3 g, if not masked by other effects. The g for both the Be, and Bee error terms. From Figure 10 it can be seen J TILT ANGLE $ FIGURE 10 MAGNITUDE AND PRECISION OF INPUTS Since both the buoyancy factor B and the angle of tilt e are servo-controlled, each can be precisely and independently varied by introducing appropriate biases with their Servo-control loops. Function generators or analog computers can be used to generate any desired input time-history, and frequency correlation of the accelerometer Output with the known inputs can result in very accurate parameter estimation. U s e of on-line computers permits complex instrument modeling and yields essentially instantaneous values for the parameters in the model. VIII. SUhlMARY The development of low-level space accelero- meters has been restricted by the difficulty of calibrating these instruments in the presence of the earth's one-g gravitational field. A few researchers have ettemptcd to Work with a true zero-g environment (free fall) ( 5 ) , but this approach is often impractical due to time and Cost limitations and the level of complexity of adequate instrumentation. Experimental results have confirmed that B simulated zero-g environment can be achieved with the flotation technique with the following advantages: The instrument under test operates a t force levels comparable to those expected in orbit. The flotation factor B attenuates external disturbances such a s Seismic noise. External inputs are relatively large , permitting great preoirion. Use of a preoision tilt table can yield effective inputs a s small as 10-13 2, . Calibration procedures are suitable for on-line computer reduction. Orbital frequencies a r e easily simulated. 68 - 874 The present flotation facility has a capability of controlling the flotation factor B to its7. figure is due partly to instrumentation limitations and partly to the design of the prototype facility. The ease with which this figure was attained indicates that future designs could possibly improve this performance to or perhaps even This 10-9. The flotation technique is not applicable to all types of accelerometers, but is a potentially Powerful method f o r calibrating those instruments Which employ low-density proof masses and a r e capable of operation in the flotation fluid. REFERENCES 1. "Low-G Accelerometers Await New Ground and Flight Test Developments," Missiles and Rockets, Dec. 7, 19b4, pp. 32-35. 2. Meldrum, M.A., Harrison, E.J., and Milbum, Z . , "Development of a Miniature Electrostatic Accelerometer (MESA) for Low-G Applications," NASA Report CR-54137, 1965. 3 . Stokes. G.G.. "On the Effect of the Internal Friction of Fluids bn the Motion of Pendulums," Cambridge Philosophical Society Transactions I X (1851), Part 2, PP. 8-106. 4. DeBra, D.B., Mathieden, J.C., and VanPatten, R.A., "A Precision, Active, Table Leveling System." paper presented at the 1966 AIAA Guidance and Control Specialist Conference, Seattle, Washington, August 1966. 5. Delattre, M.,,"L'Acceierometre 0.N.F.R.A. A Grande Sensihilite." Office National D'EtudeS et , . de Recherches A&-Sbatiales (France) Report T.P.n-449, 1967. 11