Analog Computers

Reference / Paper · 1966

Stability Research on Parachutes Using Digital and Analog Computers

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NASA technical translation (TT F-10,391) of a 1963 German conference paper by R. Ludwig, presented at the International Symposium on Analog and Digital Techniques Applied to Aeronautics in Liege, Belgium. The paper investigates the dynamic stability of parachutes using nonlinear equations of motion solved on both digital computers (IBM 650, Siemens 2002, using Runge-Kutta) and an analog computer, covering approximately 80 computed cases. It demonstrates that linearization is fundamentally inadequate for describing parachute oscillation dynamics and that analog and digital computation together are essential for this class of nonlinear flight mechanics problems.

Manufacturer
NASA / Deutsche Forschungsanstalt fur Luft- und Raumfahrt
Author
R. Ludwig
Year
1966
Type
Reference / Paper
Language
English
Learning track
specific applications
Pages
20
  • NASA / Deutsche Forschungsanstalt fur Luft- und Raumfahrt
  • parachute stability
  • nonlinear differential equations
  • analog computation
  • flight mechanics

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Stability Research on Parachutes Using Digital and Analog Computers

? NASA TT F-10,391 GPO PRICE $ - CFSTI PRICE(S1 $ Hard copy IHC) 9 ),OD Microfiche (MF) I 5 6 n 663 July 66 STABILITY RESEARCH ON PARACHUTES USING DIGITAL AND ANALOG COMPUTERS R. Ludwig Translation of: "Stabilitatsuntersuchungen an Fallschirmen mit Hilfe eines Digital- und Analogrechners." Paper presented at the International Symposium on Analog and Digital Techniques Applied to Aeronautics , LiGge, Belgium, Sept 9-12, 1963 (13 pp. and 6 Illus.). Deutsche Forschungsanstalt fcr Luft- und Raumfahrt E.V. , Braunschweig, 1963. . N 0 d-v 5: m 0 L -4> (PAGES1 < L (NASA CR OR TMX OR A D NUMBER1 * e (THRU) I 37 (CATEGORY) . NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON NOVEMBER 1966 . NASA TT F-10,391 c' STABILITY RESEARCH ON PARACHUTES USING D I G I T A L AND ANALOG COMPUTERS R. Ludwig The computation of numerous examples of a s p e c i a l t y p e ( n u m e r i c a l l y about 80 cases were c o n s i d e r e d ) shows t h a t t h e o s c i l l a t i o n s of a p a r a c h u t e show a c e r t a i n t y p i c a l t y p e of b e h a v i o r which i s c h a r a c t e r i s t i c of n o n l i n e a r o s c i l l a t i o n s . A q u a l i t a t i v e agreement w i t h experiments w a s achieved i n a number of r e s p e c t s . Q u a n t i t a t i v e comparative i n v e s t i g a t i o n s s t i l l could n o t b e c a r r i e d o u t because u n t i l now i t s t i l l w a s n o t p o s s i b l e t o c a r r y o u t drop experiments w i t h c h u t e s of t h e c o n s i d e r e d t y p e . I n a d d i t i o n t o t h e i n f o r m a t i o n which t h e experimenter o b t a i n s on d i f f e r e n t p r o p e r t i e s of p a r a c h u t e o s c i l l a t i o n , i t a p p e a r s t o b e p a r t i c u l a r l y important t h a t t h i s i s a case where f o r t h e i n v e s t i g a t i o n of t h e dynamic b e h a v i o r a n o n l i n e a r computation i s t h e o n l y approach which can g i v e a n unobject i o n a b l e d e s c r i p t i o n of t h e process. The a v a i l a b l e e x p e r i mental d a t a , such as wind t u n n e l measurements f o r asymmetrical c h u t e s i n 6 components, measurements of t h e e n t r a i n e d a i r m a s s , etc., s h o u l d b e used i n f u r t h e r broadening of theoretical investigations. 1. Introduction /1* I n t h e c o n s i d e r a t i o n of dynamic problems i n f l i g h t mechanics, i t w a s customary a t a n earlier t i m e t o l i n e a r i z e t h e problem, t h a t i s , t h e e f f e c t of s m a l l p e r t u r b a t i o n s w a s c o n s i d e r e d . The system of l i n e a r d i f f e r e n t i a l equat i o n s f o l l o w i n g from t h i s approach a l s o had t h e p l e a s a n t p r o p e r t y t h a t w i t h a r e l a t i v e l y minor number of computations it w a s p o s s i b l e t o draw c o n c l u s i o n s concerning f r e q u e n c i e s and a t t e n u a t i o n s . The a d m i s s i b i l i t y of l i n e a r i z a t i o n i n many cases w a s q u e s t i o n a b l e from t h e beginning. Now, on t h e o t h e r hand, i n most cases t h e r e h a s been a changeover t o n o n l i n e a r computations. It i s a c c e p t ed, t h u s , t h a t t h e volume of computations w i l l b e v e r y g r e a t l y i n c r e a s e d and t h a t it s c a r c e l y i s p o s s i b l e t o draw any g e n e r a l c o n c l u s i o n s ; c o n c l u s i o n s can b e drawn o n l y from numerous examples i n which c e r t a i n p a r a m e t e r s are v a r i e d . Only by u s e of t h e modern t o o l s of analog and d i g i t a l computers h a s i t become p o s s i b l e t o compute t h e dynamic problems of f l i g h t mechanics i n t h i s u n i v e r s a l ity. I n t h e i n v e s t i g a t i o n of t h e dynamic s t a b i l i t y of p a r a c h u t e s , which w i l l b e *Bumbers i n t h e margin i n d i c a t e p a g i n a t i o n i n t h e o r i g i n a l f o r e i g n t e x t . NASA TT F-10,391 b d i s c u s s e d h e r e , u n t i l now work always has begun w i t h l i n e a r i z e d e q u a t i o n s of motion. Even though t h e f i r s t p u b l i c a t i o n known t o t h e a u t h o r had a l r e a d y appeared i n 1918 [ l ] , a communication published by W. G. S. Lester (1962) [ 3 ] made no mention of i t . A s w i l l b e assumed h e r e i n advance, t h e r e s u l t of l i n e a r i z a t i o n i n t h i s case i s p a r t i c u l a r l y complicated. The d i f f e r e n t i a l e q u a t i o n f o r t h e p e r t u r b a t i o n of v e l o c i t y i s s p l i t o f f from t h e o t h e r d i f f e r e n t i a l equat i o n s and g i v e s a monotonic a t t e n u a t i o n of a p e r t u r b a t i o n , b u t n o t a p e r i o d i c a t t e n u a t i o n w i t h t h e frequency of t h e o s c i l l a t i n g c h u t e . Here, t h i s means t h a t l i n e a r i z a t i o n , r e g a r d l e s s of whether t h e a p p l i c a t i o n of t h e t h e o r y of s m a l l o s c i l l a t i o n s i s a d m i s s i b l e , l e a d s t o a d e f i c i e n t d e s c r i p t i o n of t h e p h y s i c a l behavior. From t h e p o i n t of view of p a r a c h u t e technology, t h e a t t a i n m e n t of good dynamic s t a b i l i t y i s of g r e a t importance. R e g a r d l e s s of t h e o b j e c t i v e of t h e c h u t e -- whether f o r s a v i n g a p i l o t , f o r e j e c t i o n , f o r b r a k i n g t h e l a n d i n g of an a i r c r a f t o r as a c h u t e f o r s t a b i l i z i n g any kind of f l i g h t v e h i c l e -- i n a l l cases an i n s u f f i c i e n t s t a b i l i t y w i l l a t l e a s t l e a d t o d i f f i c u l t i e s o r even de-& s t r o y t h e real purpose of t h e c h u t e . Moreover, t h e assumption of s m a l l p e r t u r b a t i o n s a l s o i s s c a r c e l y r e a l i z e d i n p r a c t i c e . The i n f l u e n c e of a g u s t on a s t a b l y f a l l i n g c h u t e v e r y e a s i l y can l e a d t o d e f l e c t i o n s which no l o n g e r j u s t i f y a l i n e a r i z a t i o n . The aerodynamic v a l u e s f o r r e s i s t a n c e ( d r a g ) , s h e a r and moment i n dependence on a n g l e of a t t a c k , needed f o r computations, as known from wind t u n n e l measurements, l i k e w i s e show a b e h a v i o r which does n o t admit a l i n e a r i z e d treatment, as i s customary i n f l i g h t mechanics, f o r example Aa because i n p a r t a cr,/ a a even varies i n s i g n . For t h e model c a l c u l a t i o n s , which w i l l be r e p o r t e d on h e r e , t h e s o - c a l l e d p e r s o n n e l g u i d e s u r f a c e p a r a c h u t e w i l l be used. For t h i s p a r a c h u t e , whose prot o t y p e w a s developed d u r i n g t h e Second World War by P r o f . H e i n r i c h a t t h e Aeron a u t i c a l I n s t i t u t e i n S t u t t g a r t , i n Germany ( P r o f . Madelung) we have American wind t u n n e l measurements which were c a r r i e d o u t a t t h e I n s t i t u t e by P r o f . Heinr i c h ( U n i v e r s i t y of Minnesota, USA). I n t h e numerous computed examples, w e i n v e s t i g a t e d t h e dependence of d i f f e r e n t p a r a m e t e r s , such as t h e i n f l u e n c e of t h e l e n g t h of t h e shroud l i n e s , the v a r i a t i o n of t h e mass of t h e e n t r a i n e d a i r , and t h e d e n s i t y of t h e s u r r o u n d i n g a i r ( o r a l t i t u d e ) . F i n a l l y , t h e s t a b l e state of o s c i l l a t i o n a l s o w a s c o n s i d e r ed f o r t h e case i n which, i n a c e r t a i n neighborhood of t h e a n g l e of attack z e r o , t h e moment i s n o t r e s t o r i n g ( t h a t i s , XM/aa v a r i e s i n s i g n i n t h e corresponding region). 2 . 2. /3 Notations [m/sec] = v e l o c i t y v e c t o r , sum of v e l o c i t y , components i n a c o o r d i n a t e syst e m r e l a t e d t o the parachute [m/sec] = s t a b l e speed of d e s c e n t [sec-ll = angular velocity [kg] = l o a d on t h e p a r a c h u t e "L 2 [ k g - s e c /m] = m a s s of t h e l o a d 2 [ k g - s e c /m] = a i r mass e n t r a i n e d by shroud [ l ] = r a t i o of t h e e n t r a i n e d a i r m a s s t o t h e m a s s of t h e l o a d [ l ] = a n g l e of i n c l i n a t i o n of t r a j e c t o r y Y [ l ] = l o n g i t u d i n a l a n g l e of i n c l i n a t i o n a [ l ] = a n g l e of a t t a c k [kgomosec 2 , m] = moment of i n e r t i a , r a d i u s of t h e shroud S [m] = d i s t a n c e from midpoint of shroud t o p o i n t of a p p l i c a t i o n of l o a d Y [kg] = i n t e r n a l f o r c e [kg] = e x t e r n a l f o r c e W [kg] = resistance Q [kgl = s h e a r M [kg-m] = moment [ l ] = r e s i s t a n c e , shear and moment of shroud 2 F=RIT 2 [m 3 = r e f e r e n c e p l a n e of p a r a c h u t e shroud f o r t h e a i r f o r c e R [m] = r a d i u s of t h e p a r a c h u t e shroud /4 3 [sec] = t i m e t [m] = c o o r d i n a t e s of t r a j e c t o r y i n a c o o r d i n a t e system r e l a t e d t o t h e ground x, Y n g [m/secL] = a c c e l e r a t i o n of g r a v i t y 2 4 [kgosec /m ] = a i r d e n s i t y 5 The t i m e d e r i v a t i v e s are denoted by a d o t . Subscripts: K = shroud; L = load. 3. Equations &Motion The e q u a t i o n s of motion w i l l o n l y be c o n s i d e r e d b r i e f l y [ 5 ] . The premises are : a) b) chute. The shroud-load system is r i g i d . The motion o c c u r s i n a v e r t i c a l p l a n e p a s s i n g through t h e a x i s of t h e c ) The shroud e n t r a i n s a n a i r m a s s which i s t o b e r e g a r d e d as a s l u g g i s h b u t n o t as a heavy mass; i t w i l l b e c a l l e d t h e a p p a r e n t ( a s i n E n g l i s h ) o r ent r a i n e d m a s s . On t h e o t h e r hand, t h e mass of t h e c h u t e can be n e g l e c t e d . d) The c h u t e is a c t e d upon by aerodynamic f o r c e s , r e s i s t a n c e i n t h e d i r e c t i o n of t h e t r a j e c t o r y and t h e s h e a r p e r p e n d i c u l a r t o i t , a n d an aerodynamic moment about an a x i s p e r p e n d i c u l a r t o t h e p l a n e of t h e t r a j e c t o r y . On t h e l o a d , t h e r e s h o u l d b e only a n e g l i g i b l y s m a l l r e s i s t a n c e , b u t no s h e a r and no moment. Now we w i l l c o n s i d e r t h e f o r c e e q u a t i o n s f o r t h e shroud and l o a d s e p a r a t e l y (Figure 1) : and t h e e q u a t i o n of moment, r e l a t e d t o L i n a c o o r d i n a t e system r e l a t e d t o t h e parachute 4 . . F i g u r e 1. N o t a t i o n s . It f o l l o w s from t h e premise of r i g i d i t y of t h e shroud-load system t h a t : 5 from t h e f i g u r e w e a l s o have t h e g e o m e t r i c a l n o t a t i o n s As t h e o p e r a t i n g e x t e r n a l f o r c e s Then w e p u t t h e aerodynamic f o r c e s and moments i n t h e u s u a l form: 16 I f w e s u b s t i t u t e e q u a t i o n s (4)-(10) i n t o e q u a t i o n s (1)-(3) and, i n addit i o n , i n t r o d u c e d i f f e r e n t i a l a e q u a t i o n s f o r t h e t r a j e c t o r y of t h e l o a d and shroud, w e o b t a i n f i f t e e n v a l u e s f o r t h e complete d e s c r i p t i o n of t h e dynamic beh a v i o r of t h e c h u t e , namely, f o r t h e motion of t h e l o a d : and f o r t h e motion of t h e shroud: 6 Vq 1 V q I ?I( I *(Y /PI<I y< We now have a system of 6 differential equations and 9 algebraic notations. 7 I n a d d i t i o n , w e have t h e i n i t i a l c o n d i t i o n s f o r t i m e t = 0. W e w i l l assume t h a t w h i l e t h e c h u t e i s i n s t a b l e v e r t i c a l motion w i t h t h e speed i t i s d e f l e c t e d l a t e r a l l y i n t h e t r a j e c t o r y p l a n e by a g u s t , o r t h e l i k e , by an a n g l e 9oand t h e n I n t h e model computations, t h e e q u a t i o n s (12) and (13) are transformed i n s u c h and & a way t h a t t h e r e w i l l b e one e q u a t i o n each f o r Y 8 4. Computation Methods The s o l u t i o n s of t h e system of 6 d i f f e r e n t i a l e q u a t i o n s were u b t a i n e d usi n g b o t h d i g i t a l and analog computers. With r e s p e c t t o t h e d i g i t a l computation we n o t e t h e f o l l o w i n g : The computations f i r s t w e r e c a r r i e d o u t on a n IBM 650 (AVA, G a t t i n g e n ) and t h i s y e a r on a Siemens 2002 of t h e DFL. T h e r e f o r e , t h e u s u a l s o l u t i o n method /8 of s t e p i n t e g r a t i o n by t h e Runge-Kutta method ( f o u r t h o r d e r ) w a s used. The p r o g r a m i n g w a s accomplished u s i n g t h e symbolic SOAP o r HAS1 programming l a n C and C were t a k e n from a t a b l e as a funcguages. The aerodynamic v a l u e s C W’ Q M t i o n of t h e a n g l e of a t t a c k aK and i n t e r p o l a t e d l i n e a r l y . S i n c e t h e i n t e r p o l a t i o n of t h e t h r e e f u n c t i o n s w i t h i n a Runge-Kutta i n t e r v a l must b e c a r r i e d o u t f o u r t i m e s , i t i s recommended t h a t t h e s e a r c h t i m e b e s h o r t e n e d u s i n g f o r t h i s purpose a s p e c i a l l y r e s e r v e d index, an i n d i c a t o r of t h e l a s t computed p l a c e soto-speak, and from t h e r e on, above and below, s e e k t h e p r o p o r t i o n a t e l y near-lyI n c e r t a i n computations extending over g r e a t e r t i m e i n t e r v a l s , t h e i n g value. v a l u e s are approximated by ( f i f t h o r s i x t h d e g r e e ) polynomials (as d i r e c t o r i n d i r e c t f u n c t i o n s ) , r e s u l t i n g i n a f u r t h e r s a v i n g of time w i t h o u t a l o s s of accuracy. By t r i a l and e r r o r , w e determined s u i t a b l e i n t e r v a l s A t f o r a t t a i n i n g t h e r e q u i r e d accuracy. I n t h e case of l o n g e r t r a j e c t o r i e s , t h e v a l u e A t = 0.05 sec w a s used, b u t o n l y each 1 0 t h s t e p w a s used. For i n c r e a s i n g c l a r i t y , and a l s o f o r s h o r t e n i n g t h e computation t i m e (at t h e t i m e o n l y t h e Siemens 2002 w i t h punch t a p e p r i n t o u t (60 symbols/sec) w a s a v a i l a b l e ) , t h e computations w e r e made, t o b e s u r e , w i t h a f l o a t i n g p o i n t ( 1 0 - d i g i t m a n t i s s a ) , b u t a l s o w i t h a f i x e d p o i n t w i t h a r e a s o n a b l e number of d e c i m a l s set a s i d e ( f o r example, i n t h e case of t r a j e c t o r y c o o r d i n a t e s i n meters -- 2 d e c i m a l s ) . Comments on Computations with t h e Analog Computer* A PACE 231 R a n a l o g computer of E l e c t r o n i c A s s o c i a t e s , Inc., w a s a v a i l a b l e . For r e d u c t i o n of m u l t i p l i c a t i o n u n i t s , t h e aerodynamic v a l u e s w e r e used i n a c o o r d i n a t e system r e l a t e d t o t h e body and a l s o were approximated i n p a r t by polynomials i f t h e dependence on c e r t a i n p a r a m e t e r s w a s under i n v e s t i g a t i o n . Here we even went s o f a r , f o r example, as t o approximate t h e e x p r e s s i o n C (%)= X C (- tanh-l V / V ) through a polynomial Cx*(V X Y X Y /V ) i n t h e p e r t i n e n t region. X The r e s u l t s o b t a i n e d w e r e a c c u r a t e t o about 1%.The r e a s o n f o r t h e s e /9 i n a c c u r a c i e s w e r e l a g e r r o r s of t h e servomechanisms ( d e s p i t e a q u i t e slow comp u t a t i o n ) . Transformation i n p o l a r c o o r d i n a t e s w i t h r e s o l v e r s h a s n o t proven i t s e l f , s i n c e a i s s u b j e c t t o o n l y minor f l u c t u a t i o n s and a t t h e same t i m e t h e K *The computations on t h e a n a l o g computer were made through t h e k i n d n e s s of H e r r Dip1.-Math. H. Hentschel. 9 l i m i t e d r e s o l v i n g power of t h e sine-cosine p o t e n t i o m e t e r becomes n o t i c e a b l e . A s a supplementary c o n d i t i o n , t h e energy e q u a t i o n can b e i n t r o d u c e d ; t h i s i s r e c e i v e d by m u l t i p l y i n g t h e v e c t o r e q u a t i o n of t h e t r a n s l a t i o n scalar by t h e v e l o c i t y v e c t o r 4 and m u l t i p l y i n g the moment e q u a t i o n by w and c a r r y i n g o u t t i m e i n t e g r a t i o n f o r b o t h . T h i s energy e q u a t i o n i n t r o d u c e d as a supplementary c o n d i t i o n w a s used f o r improving t h e computations u s i n g t h e s t e e p e s t d e s c e n t method. Here a l s o , as a r e s u l t of d i f f e r e n t i a t i o n f o r k, t h e v a l u e i s approximated by a polynomial. Now S = E 2 I f E i s t h e energy, t h e n we w i l l have must b e minimized. Then w e w i l l have and with A being an amplification f a c t o r . w e t h e n have: For t h e system of d i f f e r e n t i a l e q u a t i o n s , t h a t i s , i t i s n e c e s s a r y t o s h i f t t o t h e z e r o p o s i t i o n s of t h e p a r t i a l d e r i v a t i v e s u s i n g a comparator. T h i s i s i n some f e a t u r e s t h e e s s e n c e of t h e s t e e p e s t d e s c e n t method. These computations could g e n e r a l l y n o t b e c a r r i e d o u t w i t h t h e analog computer a t o u r d i s p o s a l because t h e o u t f i t t i n g w i t h components d i d n o t s u f f i c e . The o p e r a t i o n w a s s i m u l a t e d d i g i t a l l y u s i n g an A l g o l program. 5. Model Computations /10 A s a l r e a d y mentioned, t h e computations w e r e c a r r i e d o u t f o r t h e s p e c i a l p e r s o n n e l g l i d e s u r f a c e p a r a c h u t e (Figure 2) and t h e f o l l o w i n g d a t a w e r e selected : 10 , -- . - -7 - . F i g u r e 2 . [Caption not V i s i b l d The moment of i n e r t i a ( r a d i u s of i n e r t i a ) can be determined i f t h e shroud i s m e n t a l l y r e p l a c e d by an e l l i p s o i d of r e v o l u t i o n of e q u a l volume and t h i s i s e n l a r g e d [ 2 ] ; t h e moments of i n e r t i a a r e f o r m a l l y known f o r a n e l l i p s o i d . F i g u r e 3 shows t h e aerodynamic v a l u e s determined from wind t u n n e l i n v e s t i g a t i o n s of models. F i g u r e 3 shows 3 cases of d i f f e r e n t p o r o s i t y ( t h e e f f e c t i v e poros i t y i s g i v e n as a dimensionless number, i n accordance w i t h t h e d a t a g i v e n by H. G. H e i n r i c h i n [5] f o r g e o m e t r i c a l l y uniform c h u t e s . I n p a r t i c u l a r , i n t h e case of t h e impermeable c h u t e (rl = 0) we see t h a t X M / k i n t h e neighborhood of c1 K = 0 i s n e g a t i v e , t h a t i s , i n t h i s r e g i o n t h e c h u t e h a s no r e s t o r i n g moment. A s a t y p i c a l r e s u l t w e w i l l show a c a s e ( F i g u r e 4 ) i n which t h e c h u t e i s s t a b l e i n t h e e n t i r e r e g i o n of a n g l e s of a t t a c k . A s e x p e c t e d , w e o b t a i n a t t e n u a t e d o s c i l l a t i o n s of a c e r t a i n frequency. That t h e v e l o c i t i e s V,VK and Vx have a double frequency i s e a s y t o understand i f t h e c h u t e i s r e g a r d e d as a pendulum. The a p p e a r i n g minor amplitudes of o s c i l l a t i o n of t h e shroud show, as a l s o can b e s e e n on t h e t r a j e c t o r y c u r v e s , t h a t t h e l o a d e s s e n t i a l l y o s c i l l a t e s a b o u t 11 i b t *i -30' -20' -10' 0' 10' 3 ' 20' 'W F i g u r e 3 . Aerodynamic Values f o r Personnel Glide Surface Parachute. t h e shroud. These v a l u e s , p l o t t e d as a f u n c t i o n of t i m e , o n l y i n t h e case of more exact s t u d y reveal d e v i a t i o n s from t h e o s c i l l a t i o n b e h a v i o r of a l i n $ a r system. T h i s becomes clearer i n a phase diagram ( F i g u r e 5) i n which w is It can b e seen c l e a r l y from t h e t i m e marks p l o tted p l o t t e d as a f u n c t i o n of 9. on the s p i r a l t h a t the d u r a t i o n of o s c i l l a t i o n d e c r e a s e s w i t h amplitude. It a l s o i s e a s y t o l e a r n from t h e amplitude r a t i o s t h a t a t t e n t u a t i o n d e c r e a s e s w i t h amplitude. =a Now w e w i l l c o n s i d e r t h e t r a j e c t o r y c u r v e s of t h e shroud and l o a d ( F i g u r e 6 ) , i n which t h e c h u t e i s sketched i n s c h e m a t i c a l l y a t 1-second i n t e r v a l s ; t h u s , w e can v a r y t h e p o r o s i t y ( a , b , c ) , or w i t h t h e s a m e p o r o s i t y w e can v a r y t h e /11 l e n g t h of t h e shroud l i n e s ( c , d , e ) . We f i n d t h a t w i t h i n c r e a s i n g p o r o s i t y , t h e a t t e n u a t i o n increases, w i t h a lesser d e c r e a s e of t h e d u r a t i o n of o s c i l l a t i o n . The d u r a t i o n of o s c i l l a t i o n i n c r e a s e d , by analogy w i t h a pendulum, w i t h the and c1 as l e n g t h of t h e shroud l i n e s . T h i s i s shown by the v a l u e s V, V f u n c t i o n s of t i m e ( F i g u r e 7 ) . Here, about 20 cases were I n v e g t i g a t e d and t h e a s t o n i s h i n g f a c t w a s d i s c o v e r e d t h a t t h e s q u a r e of t h e d u r a t i o n of o s c i l l a t i o n - p r o p o r t i o n a l to t h e shroud-load d i s t a n c e . T h i s s u g g e s t s t h e p o s s i b i l i t y of is ,a 12 4 mise 0 -4 0,4 0 -q4 44 0 . F i g u r e 4 . Example: Temporal Variat i o n ; rl = 0.096, s = 9.1 m y \90 = 0.25. r e p r e s e n t i n g t h e observed f a c t s i n an empirical formula, u s i n g t h e formula f o r a mathematical pendulum, supplemented by a p r o p o r t i o n a l i t y f a c t o r . I f w e reby t h e s t a b l e v e l o c i t y of d e s c e n t vs, we p l a c e t h e r e s i s t a n c e (drag) value $0 obtain The o s c i l l a t i o n d u r a t i o n s computed u s i n g t h i s formula a g r e e w e l l w i t h t h e model computations ( F i g u r e 7,b). The a t t e n u a t i o n i s i n f l u e n c e d t o only a modest e x t e n t by change of t h e l e n g t h of t h e shroud l i n e s . In t h e c a s e of a s t a b l e c h u t e , t h e r e w i l l b e a weak maximum i n t h e r e g i o n of shroud l i n e s of o r d i n a r y l e n g t h . The assumption concerning t h e e n t r a i n e d a i r mass r e q u i r e s f u r t h e r checking. The computations c a r r i e d o u t ( F i g u r e 8 ) f o r d i f f e r e n t mass r a t i o s pK w i t h a cons t a n t l o a d show t h a t t h e d u r a t i o n of o s c i l l a t i o n i s v i r t u a l l y independent of 13 .. i F i g u r e 5. Example: Phase Diagram; GL = 100 kg, rl = 0.096, = 0.25. a0 t h e n t r a i n d a i r m a s s . The a t t e n u a t i o n i n d e e d i s somewhat g r e a t e r i n t h e case of a s m a l l e r mass, b u t t h e i n i t i a l d e f l e c t i o n of t h e shroud a l s o i s i n i t i a l l y enlarged. 7 I n a l l of t h e c a s e s c o n s i d e r e d u n t i l now, t h e a i r d e n s i t y h a s been asFor t h e motion of a parasumed c o n s t a n t , e q u a l t o t h a t a t t h e ground = 7 To. c h u t e at o t h e r a i r d e n s i t i e s , t h a t is, f o r o t h e r a l t i t u d e s , t h e f o l l o w i n g assumptions can be made: a) t h e aerodynamic v a l u e s remain unchanged ( t h a t is, p o r o s i t y does n o t change); b) t h e e n t r a i n e d a i r m a s s should d e c r e a s e i n mass i n such a way t h a t t h e ent r a i n e d a i r volume r e ya.n s unchanged, t h a t is I n computing a p a r t of t h e t r a j e c t o r y , t h e a i r d e n s i t y a g a i n i s h e l d 14 /12 F i g u r e 6. Example: T r a j e c t o r y Curves. Bo = 0.25; a ) s = 9.1 m y TI = 0; b) s = 9 . 1 m y TI = 0.042; c ) s = 9.1 m y q = 0.096; d ) s = 5.1 my q = 0.096; e) s = 13.1 m y TI = 0.096. c o n s t a n t . The computations show ( F i g u r e 9 ) t h a t t h e l a t e r a l d e f l e c t i o n of t h e shroud is g r e a t e r t h a n a t t h e ground, t h e a t t e n u a t i o n i s somewhat g r e a t e r and t h e d u r a t i o n of o s c i l l a t i o n i s less. I f w e compare t h e t r a j e c t o r y c u r v e s f o r d i f f e r e n t cases i n b o t h s t a b l e and u n s t a b l e cases ( F i g u r e l o ) , t h e i n i t i a l c o n d i t i o n s are d e c i s i v e f o r t h e o s c i l In an u n s t a b l e l a t i o n b e h a v i o r . The s t a b l e c h u t e h a s a v e r t i c a l t r a j e c t o r y . case, w i t h a small i n i t i a l d e f l e c t i o n , t h e c h u t e i s d e f l e c t e d f u r t h e r . T h i s r e s u l t s i n a motion i n which a l a t e r a l v e l o c i t y component w i l l b e m a i n t a i n e d , t h a t i s , t h e c h u t e i s d r i v e n sideways. I f , perchance, w e s t u d y VKy, aK and VK, we see ( F i g u r e 11) t h a t , i n gen- eral, an a t t e n u a t e d o s c i l l a t i o n a p p e a r s , b u t a s t a b l e c o n d i t i o n of o s c i l l a t i o n can b e a t t a i n e d o n l y i f t h e c h u t e h a s reached a p o s i t i o n i n which a C / a a i s M K p o s i t i v e . I n t h i s case, t h e c o u r s e of motion w a s followed o v e r 200 seconds and, of c o u r s e , t h e a i r d e n s i t y a l s o w a s h e l d c o n s t a n t h e r e i n o r d e r n o t t o v a r y s t i l l a n o t h e r parameter. The p o s i t i o n for which aCM/aaK = 0 l i e s a t aK = 0.2. S i n c e t h e l a t e r a l v e l o c i t y becomes c o n s t a n t , t h e motion becomes r e c t i l i n e a r a t a c e r t a i n a n g l e t o t h e v e r t i c a l (about 20'). 15 . . . . k -.-.- qlm 5,lm 4096 :O,096 F i g u r e 7. Example: Duration of O s c i l l a t i o n and Attenua t i o n . Legend: a = On t h e Basis of t h e E m p i r i c a l Formula. F i g u r e 8. Example: Angle of I n c l i n a t i o n of T r a j e c t o r y f a r D i f f e r - 1.0; -.-pK = 1.4. e n t E n t r a i n e d A i r Masses; ---vK = 0.6; - p K 16 " Figure 9. Example: Angle of Inclination of Trajectory for Different Altitudes. -H = 0 km, vK = 1-00;--- H = 2 km, vK = 0.822; - - H = 6 km, vK = 0.538; ... H = 10 km, pK = 0 . 3 3 8 . 6. Summary The computation of numerous examples of a special type (numerically about 80 cases were considered) shows that the oscillations of a parachute show a certain typical type of behavior which is characteristic of nonlinear oscillations. A qualitative agreement with experiments was achieved in a number of respects. Quantitative comparative investigations still could not be carried out because until now it still was not possible to carry out drop experiments with chutes of the considered type. In addition to the information which the experimenter obtains on dif/13 ferent properties of parachute oscillation, it appears to be particularly important that this is a case where for the investigation of the dynamic behavior a nonlinear computation is the only approach which can give an unobjectionable description of the process. The available experimental data, such as wind tunnel measurements for asymmetrical chutes in 6 components, measurements of the entrained air mass, etc., should be used in further broadening of theoretical investigations. 17 jectory Curves. I t -- -_18 NASA TT F-10,391 REFERENCES 1. Brodetzky, S . : The stability of the parachute. 14: 116-123, 1918- The Tohoku Math. Journ. 2. Henn: Die Absinkeiaenschaften van Fallschirmen. (The Drop Properties of Parachutes.) Zentrale fiir Wiss. Berichtwesen, Berlin-Adlershof, TJM 6202, 1944. 3. Lester, W. G. S . : A Note of the Theory of Parachute Stability. craft Establishment TN No. Mech. Eng. 358, 1962. 4. Heinrich, H. G. and E. L. Haak: Varying Effective Porosity. 1962. 5. Royal Air- Stability and Dragof Parachutes with Aeronautical Systems Division, TDR-62-100, Ludwig, R. and W. Heins: Investigations on the Dynamic Stability of Personnel Guide Surface Parachutes. DFL Bericht No. 203, Braunschweig, 1963. FRANK C. FARNHAM COMPANY 133 South 36th Street Philadelphia, Pa. 19104 19