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Biomechanics of swimming in the frog, Hymenochirus boettgeri Gál, Julianna Mary 1987

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BIOMECHANICS OF SWIMMING IN THE FROG, HYMENOCHIRUS BOETTGERI By JULIANNA MARY GAL B.Sc. The U n i v e r s i t y of B r i t i s h Columbia, 1984. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1987 © J u l i a n n a Mary G a l , 1987. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6(3/81) i i ABSTRACT Although frogs are recognized as accomplished swimmers, no d e t a i l e d biomechanical study has been done. The hydrodynamics and mechanics of swimming, i n the f rog , Hymenochiius boettgezi, are inves t iga ted i n t h i s t h e s i s . Hydrodynamic drag , of the body and splayed hind limbs of preserved H. b o e t t g e r i , was assessed by drop-tank experiments. Drag tes t s were a l so performed with the s e m i - t e r r e s t r i a l Rana p i p i e n s . A comparison of t h e i r drag c o e f f i c i e n t s (C n ) under dynamical ly s i m i l a r c o n d i t i o n s , suggests that jumping performance may not compromise the swimming a b i l i t y of R. p i p i e n s . Drag of the expanded foot of H. b o e t t g e r i , and acetate models thereof , was inves t iga ted by free f a l l drop-tank experiments, and a sub t rac t ion technique. The r e s u l t s of these methods and flow v i s u a l i z a t i o n experiments support the assumption that animal paddles can be t reated as three dimensional f l a t p l a t e s , or iented normal to the d i r e c t i o n of f low. Cine f i lms were used to study swimming during the n e a r - v e r t i c a l breathing excursions of H. b o e t t g e r i . The a c c e l e r a t i o n of frogs throughout hind limb extension (power s t r o k e ) , i s d i s t i n c t from other drag-based paddlers (eg. ange l f i sh and water boatman), which acce lera te and decelerate wi th in the power stroke phase. The propul s ive force generated dur ing the power stroke of a s i n g l e sequence (sequence 1) i s c a l cu l a ted from quasi-steady drag ( s ta t ic-body drag measurements, see Chapter I) and i n e r t i a l cons idera t ions . A d d i t i o n a l components of the force i i i balance, including the net effect of gravity and buoyancy, and the longitudinal added mass forces associated with the frog's body, are integrated to establish upper and lower bounds of the propulsive force. The propulsive force remains positive throughout extension. The val i d i t y of using static drag estimates to describe dynamic resistance is explored. Results from Chapter II suggest that simple drag-based models may not be sufficient to explain the swimming patterns observed. The right hind limb of the sequence 1 animal was modelled as a series of linked circular cylinders (the femur, tibiofibula, and metatarsal-phalangeal segments) and a f l a t plate (the foot). A blade-element approach was used to calculate the instantaneous drag-based and accelerative force components (parallel to the direction of motion) generated by hind limb flexion and extension. The negative thrust, generated by hind limb flexion, is probably responsible for the observed deceleration of the sequence 1 animal. Positive thrust is generated only during the i n i t i a l stages of extension, almost exclusively by the feet. The impulse of the accelerative-based thrust far exceedes the impulse of the drag-based thrust. Negative thrust is initiated midway, and continues thoughout extension, despite the acceleration of the animal. Hind limb interaction, is thought to provide propulsive thrust for the latter half of the extension phase. A jet and/or ground effect may be involved. It is suggested that a combination of reactive, resistive and interactive forces are required to explain propulsion in H. boettgeri, and probably other anurans. i v TABLE OF CONTENTS Abstract i i Table of Contents i v L i s t of Tables v i L i s t of F igures v i i L i s t of Symbols x Acknowledgements x i i i General Introduct ion 1 Chapter One: Hydrodynamic Drag of Hymenochirus boettgezi and Rana p ip iens Abstract 3 Introduct ion 4 Mater ia l s and Methods 6 Results 10 Discuss ion 22 Chapter Two: Whole Body Kinemat ics : Motion Ana lys i s and Es t imat ion of the Propul s ive Force Generated by the Frog , Hymenochirus boet tger i Abstract 26 Introduct ion 28 Mater i a l s and Methods 31 Results 40 Discuss ion 47 Chapter Three : Hind Limb Kinematics : A Blade-Element Approach to C a l c u l a t i n g the Forces Generated i n F l e x i o n and Extension of the Hind Limbs of Hymenochirus boet tger i V A b s t r a c t 57 I n t r o d u c t i o n 59 Analyses 62 R e s u l t s 74 D i s c u s s i o n 83 Concluding Remarks 92 L i t e r a t u r e C i t e d 95 LIST OF TABLES Table 3.1 Hind Limb Element Morphometries of H. boettgezi v i i LIST OF FIGURES Figure 1.1: Drag force on the body and hind limbs (feet removed) 11 of H. boettgezi, p lo t t ed against v e l o c i t y i n free f a l l . F igure 1.2: Drag force on the body and hind limbs (feet removed) 12 of Rana p i p i e n s , p lo t ted against v e l o c i t y i n free f a l l . F igure 1.3: Drag c o e f f i c i e n t s of the body and hind limbs (feet 13 removed) of H. boet tger i and R. p i p i e n s , p lo t ted against Reynolds number. Figure 1.4: Drag force on the foot of H. boet tger i (oriented 14 normal to the f low) , p lo t ted against v e l o c i t y in free f a l l . Figure 1.5: Drag c o e f f i c i e n t s of the r e a l and model foot of H 15 b o e t t g e r i , p lo t t ed against Reynolds number. Figure 1.6: C a l c u l a t i n g the drag force on the feet of 16 H. boet tger i by a subt rac t ion method. Figure 1.7: Comparing the drag c o e f f i c i e n t s of the foot of 17 H. b o e t t g e r i , c a l c u l a t e d by free f a l l drop-tank experiments and a sub t rac t ion method. Figure 1.8: C a l c u l a t i n g the drag force on model feet of H 18 b o e t t g e r i , by a sub t rac t ion method. Figure 1.9: Comparing the experimental drag c o e f f i c i e n t s for the 19 r e a l and model foot of H. b o e t t g e r i , with l i t e r a t u r e values for t e c h n i c a l equ iva lent s . v i i i F igure 1.10: The flow c h a r a c t e r i s t i c s of the body and hind limbs 21 (feet removed) of ff. b o e t t g e r i , i n free f a l l . F igure 2 .1 : The Locam camera speed curve 32 Figure 2.2: The camera and tank set-up for f i l m i n g the breathing 33 excursions of H. b o e t t g e r i . Figure 2 .3 : Sca l ing the t o t a l wetted surface area with snout-vent 37 length i n H. b o e t t g e r i . Figure 2.4 : Sca l ing the t o t a l body mass with snout-vent length i n 38 H. boettgezl. Figure 2 .5 : Smooth and experimental whole body kinematics (vent 41 displacement, v e l o c i t y , a c c e l e r a t i o n , and snout-vent length) of H. b o e t t g e r i , sequence 1. Figure 2 .6 : Smooth vent v e l o c i t y of H. boettqeiif sequences 43 1-4. F igure 2 .7 : Smooth vent a c c e l e r a t i o n of H. boettgeri, sequences 44 1-4. Figure 2 .8 : Comparing the drag c o e f f i c i e n t s of the s t a t i c body 46 and hind limbs (Chapter I ) , with the drag c o e f f i c i e n t s c a l c u l a t e d from the force corresponding to the d e c e l e r a t i o n of H. boet tger i dur ing hind limb f l e x i o n (sequence 1) . Figure 2 .9 : Comparing the body v e l o c i t y patterns of frogs with 48 other paddl ing animals . Figure 2.10: Es t imat ing the upper and lower bounds of the 50 propuls ive thrust of H. b o e t t g e r i , from the force balance of sequence 1. ix Figure 3.1: Whole body t r a c i n g of H. b o e t t g e r i , i l l u s t r a t i n g the 63 est imat ion of the c e n t r a l long axes of the torso and hind limb segments. F igure 3.2: Diagrammatic representat ion of the p o s i t i o n a l angle 64 of the hind limb segments, and the numbering of the hind limb elements of H. b o e t t g e r i . F igure 3 .3 : C a l c u l a t i n g the drag-based thrust of an element of 66 the r i g h t hind limb of H. boettgeri. Figure 3.4: C a l c u l a t i n g the a c c e l e r a t i v e force of an element of 71 the r i g h t foot of H. b o e t t g e r i . F igure 3.5: A symmetric composite s t i c k diagram of H. b o e t t g e r i , 75 i l l u s t r a t i n g hind limb f l e x i o n and extens ion. Figure 3.6: Experimental and smooth p o s i t i o n a l angles of the 76 r i g h t hind limb segments of H. boettgeri (sequence 1) . F igure 3.7: Smooth angular v e l o c i t i e s of the r i g h t hind limb 78 segments of H. boet tger i (sequence 1) . F igure 3.8: Smooth angular acce l e ra t ions of the r i g h t hind limb 79 segments of H. boet tger i (sequence 1 ) . F igure 3 .9 : The r e l a t i v e v e l o c i t y of the r i g h t hind limb elements 80 of H. boettgeri (sequence 1) . F igure 3.10: The t o t a l c a l cu l a ted force (drag-based and 81 a c c e l e r a t i v e ) , generated throughout hind limb f l e x i o n and extension of H. boet tger i (sequence 1 ) . F igure 3.11: Comparing the t o t a l c a l cu l a ted force (drag-based and 88 a c c e l e r a t i v e ) , with the i n e r t i a l force of the body, and the propuls ive estimate from the force balance, of H. boet tger i (Chapter I I , sequence 1) . X LIST OF SYMBOLS LENGTH r d is tance from segment p ivot point to element midpoint X snout-vent length of frog AREA S w t o t a l wetted surface area ; planform area (foot only) TIME t dura t ion of the power stroke (hind limb extension) VELOCITY (angular) w angular v e l o c i t y of a segment VELOCITY ( l i n e a r ) U terminal v e l o c i t y of a body v r e l a t i v e v e l o c i t y of an element to the f l u i d V R normal component of the r e l a t i v e v e l o c i t y of an element v spanwise component of the r e l a t i v e v e l o c i t y of an element Vg v e l o c i t y of the f rog ' s body V £ r e l a t i v e v e l o c i t y of the p ivot point of a segment ACCELERATION (angular) a angular a c c e l e r a t i o n of a segment ACCELERATION ( l i n e a r ) a a c c e l e r a t i o n of the f rog ' s body A add component of A R p a r a l l e l to the d i r e c t i o n of motion 2 A centrepeta l a c c e l e r a t i o n of an element. (ru>) / r cen A n e j . r e l a t i v e a c c e l e r a t i o n of the p ivot point of a segment A a c c e l e r a t i o n component of A , normal to the element n r e i A , vector a d d i t i o n / s u b t r a c t i o n of A . and A r e i net res x i res r e su l t an t a c c e l e r a t i o n of A and A. cen tan A. t angent i a l a c c e l e r a t i o n of an element, rot tan MASS, DENSITY mass of the frog v i r t u a l mass of an element (element mass plus added mass) f l u i d dens i ty dens i ty of the preserved frog ( t i s sue dens i ty) m m v i r t u a l P p' FORCES D dF v dT dT. . to t F 'add F addto t net T T V a V s ANGLES Y e o p Q s T drag drag corresponding to the r e l a t i v e v e l o c i t y of an element component of d F y p a r a l l e l to the d i r e c t i o n of motion t o t a l dT components of a l l elements added mass force of an element t o t a l added mass force of a l l elements (of the foot) force associated with a c c e l e r a t i n g the f rog ' s body thrus t produced during hind limb extension average thrus t produced dur ing hind limb extension weight of the preserved frog In a i r submerged weight of the preserved frog p o s i t i o n a l angle of a segment hydromechanical angle of attack of an element 9 0 ° - ^ ° (rr/2rad-yrad) 9 0 ° - S ° - T ° (Ti /2rad-Srad-Trad) r + T arcs i n ( ( A n e f . s i n Q ) / A r e l ) a r c s in (A . / A ) tan res X 1 8 0 OV ) - e ° (nrad-rrad-erad) COEFFICIENTS AND CONSTANTS Cp drag c o e f f i c i e n t Cp-dynamic drag c o e f f i c i e n t based on d e c e l e r a t i o n of the body C p - s t a t i c drag c o e f f i c i e n t based on the drop-tank experiments R g Reynolds number i a r b i t r a r y hind limb element j a r b i t r a r y hind limb segment K constant to c a l c u l a t e C of f l a t p la te ( 4 0 ° > © > 1 3 0 ° ) x i i i ACKNOWLEDGEMENTS I would l i k e to thank my research superv i sor , Dr. R. W. B lake , p a r t i c u l a r l y for encouraging me to write my f i r s t s c i e n t i f i c paper. Spec ia l thanks to Dr. J . M. Gos l ine for h i s e n t h u s i a s t i c advice throughout t h i s endeavor. I t i s a pleasure to thank Dr . D. J . Randal l for h i s continued support . Thanks a l so to Mrigesh Kshatry ia for h i s ind i spens ib le computing s k i l l s , and patience i n teaching them to me. Inasmuch as t h i s the s i s represents the culminat ion of my own e f f o r t s , col league i n t e r a c t i o n was an important factor i n the a s s i m i l a t i o n and refinement of t h i s work. I am gra te fu l to a l l of my coworkers i n comparative biomechanics (Sept . I 84-June , 87) for t h e i r c o l l e c t i v e encouragement. 1 GENERAL INTRODUCTION Aquatic propul s ion i s e s s e n t i a l l y a phenomenon of momentum conserva t ion . Swimming animals are prope l led forward by the reac t ion associated with the rearward momentum that they impart to the f l u i d . Because the experimental measurement of f l u i d momentum i s genera l ly d i f f i c u l t , the thrust generated by swimming animals must be i n f e r r e d from kinematic analyses and hydromechanical mode l l ing . This the s i s deals with the physics of aquat ic propuls ion i n Hymenochirus b o e t t g e r i , a smal l p i p i d f r o g . In Chapter I, the drag of the body and hind limbs i s measured exper imenta l ly . Drag forces r e s i s t motion wi th in f l u i d s and most aquat ic animals have evolved body morphologies that minimize i t s e f f e c t s . The drag c o e f f i c i e n t s of H. boet tger i and the t e r r e s t r i a l jumper Rana p ip iens are c a l c u l a t e d and compared to determine whether jumping performance compromises swimming a b i l i t y i n the l a t t e r . Chapter II combines the re s i s t ance measurements of Chapter I , with whole-body kinematic data der ived from high speed c ine f i l m s , to charac te r i ze the types of forces ac t ing dur ing swimming. Drag i s p r o p o r t i o n a l to the f rog ' s v e l o c i t y squared. Any a c c e l e r a t i o n ind ica te s an i n e r t i a l input . Because these animals are nega t ive ly buoyant, the net e f fec t of g r a v i t y and buoyancy i s a l so cons idered . The t o t a l propuls ive thrus t generated by the animal must be s u f f i c i e n t to overcome the r e s i s t i n g forces , s a t i s f y i n g the force requirements of the observed movement pa t te rns . Much of the d i s c u s s i o n Is concerned with the v a l i d i t y of employing r i g id -body 2 drag estimates to determine the drag force on f l e x i n g systems. Frogs swim by synchronous hind limb k i c k s ( e x t e n s i o n s ) . I t has been suggested that the drag forces a s s o c i a t e d with these movements are the source of p r o p u l s i v e t h r u s t f o r these animals. In Chapter I I I , t h i s i s t e s t e d by modelling the hind limb as a s e r i e s of l i n k e d c y l i n d e r s and a f l a t p l a t e . The t o t a l drag force i s c a l c u l a t e d from known r e s i s t a n c e p r o p e r t i e s , and the d e t a i l e d angular and l i n e a r hind limb kinematics. The estimate of t o t a l t h r u s t , i s compared to the r e s u l t s of Chapter I I . This comparison challenges the e s t a b l i s h e d assumptions regarding t h r u s t production by frogs and other paddling organisms. Each chapter i s w r i t t e n i n the format of s e l f - c o n t a i n e d s c i e n t i f i c paper. The f i n a l s e c t i o n e n t i t l e d Concluding Remarks inclu d e s h i g h l i g h t s of the previous chapters, and suggestions f o r f u r t h e r r e search. 3 CHAPTER I HYDRODYNAMIC DRAG OF HYMENOCHIRUS BOETTGERI AND RANA PIPIENS ABSTRACT Drag of the frog, Hymenochizus boettgeri was investigated by a series of drop-tank and flow visualization experiments. The drag coefficient (CD) of the body and splayed hind limbs was 0.24-0.11, for 1500<Re<8000. Results of the flow visualization experiments support the C D values obtained for the body and hind limbs of H. boettgeri. Cp similarily measured for Rana pipiens, was 0.060-0.050, for 16600<Rg<40400. A; comparison of Cp under dynamically similar conditions suggests that jumping may not compromise swimming performance in Rana pipiens. for the foot was examined by three methods, drop-tank experiments with isolated frog's feet, with isolated acetate model feet, and by a subtraction method. C n for the isolated foot was 2.5-1.6 for 100<R <700. Results are similar to those obtained with e isolated model feet, where C_ was 1.8-1.2 for 300<R <1300. The ' D e subtraction method gives similar results to those obtained from drop-tank experiments with isolated model and real feet, within the Rg range of 300-3000. The results of a l l three methods and flow visualization experiments, support the assumption that animal paddles can be treated as three-dimensional flat plates, oriented normal to the direction of flow. 4 INTRODUCTION Swimming animals generate t h r u s t by four b a s i c mechanisms; undulatory waves, drag-based p a d d l i n g , l i f t - b a s e d methods, and j e t p r o p u l s i o n . Research has focussed on undulatory body wave swimming i n f i s h , (see Wu 1971, L i g h t h i l l 1975, Webb 1975, and Blake 1983 f o r re v i e w s ) , which i s now r e l a t i v e l y w e l l understood. A wide v a r i e t y of animals swim by a pad d l i n g mechanism (eg. c e r t a i n i n s e c t s and t h e i r l a r v a e , f i s h , amphibians, b i r d s and mammals); however, few have been s t u d i e d . N a c h t i g a l l (1960, 1961, 1977, 1980) i n v e s t i g a t e d water b e e t l e swimming, f o c u s s i n g on the hydrodynamic p r o p e r t i e s of t h e i r body and swimming l e g s . Blake (1979, 1980, 1981a) developed a hydromechanical model which has been a p p l i e d to p e c t o r a l f i n pad d l i n g i n a n g e l f i s h (Blake 1979, 1980), s u r f a c e p a d d l i n g i n muskrat ( F i s h 1984), and waterboatman swimming (Blake 1986). I t i s thought t h a t anurans swim by a drag-based mechanism, wherein the rearward movement of t h e i r l a r g e p a d d l e - l i k e f e e t generate the r e s i s t i v e f o r c e s t h a t p r o p e l l the animal. However, locomotor s t u d i e s on anurans have concentrated on jumping (Rand 1966, Zug 1972, Calow and Alexander 1973, Emerson 1978, Hirano and Rome 1984). Here, the drag of Hymenochizus b o e t t g e r i i s compared t o tha t of the more t e r r e s t r i a l Rana p i p i e n s . R e s u l t s are di s c u s s e d i n r e l a t i o n t o p o s s i b l e d e s i g n compromises f o r swimming and jumping i n R. p i p i e n s . U n l i k e R. p i p i e n s , H. b o e t t g e r i i s almost wholly a q u a t i c , showing some s p e c i a l i z e d a q u a t i c a d a p t a t i o n s . For 5 example, the lateral line system, characteristic of fish and aquatic tailed amphibians, is retained (Fraser 1973). Argueably, swimming ability (as reflected by body drag coefficient) should not be compromised by design "trade offs" associated with good jumping performance in H. boettgeri, and this should be reflected in a lower drag coefficient relative to R. pipiens under dynamically similar conditions. In addition, the drag of the propulsive foot of H. boettgeri is examined in detail. Discussion focusses on the validity of employing force coefficients for man-made flat plates to characterize the drag properties of animal paddles. 6 MATERIALS AND METHODS Animals (H. boettgeri (Pipidae), weight and snout-vent length; 0.15-0.30g. and 1.5-2.5 cm. respectively, and R. pipiens (Ranidae) weight and snout-vent length; 20.Og and 7.5cm. respectively) were preserved in either a 10% formaldehyde or a 90% ethanol solution. Droptank experiments were conducted with splayed-limbed/openfoot and splayed-limbed/footless specimens of H. boettgeri, and a splayed-limbed/footless specimen of R. pipiens. Femoral and tibiofibular segments of the hind limbs were approximately orthogonal and parallel to the long axis of the body (80°-110° and 130°-180° respectively). The angle of the metatarsal -phalangeal segment ranged from 110°-180°. These angles: correspond approximately to those of the hind limb segments at the beginning of the power stroke (hind limb extension) of H. boettgeri. The 'open' foot was oriented normal to the direction of the flow. Feet were amputated at the distal end of the metatarsal-phalangeal segment. The data collected from drop-tests with the footless animals represents the drag experienced by the body and hind legs of the frog. Drop-tests were conducted with isolated real and model feet of H. boettgeri, to assess the drag of the feet alone. Additionally, the drag of the feet was estimated by a subtraction method. In drop-tank experiments with H. boettgeri, specimens f e l l freely through water to the bottom of a 10.0cm. x 10.0cm. x 50.0cm. clear plexiglass tank. The tank was lined with a 1.0cm. x 1.0cm. grid. For R. pipiens, drop-tests were conducted in a 30cm. x 30cm. 7 x 130.0cm. c l e a r g l a s s tank l i n e d with a 5.0cm. x 5.0cm. g r i d . At ter m i n a l v e l o c i t y , the submerged weight of each specimen Wg i s equal to the drag f o r c e , D: V=D=l/2pS U 2C. (1.1) s w u where p , S^, U, and C n are f l u i d d e n s i t y , the wetted surface area ( t o t a l wetted surface area f o r body and hind limbs, and planform area f or the r e a l and model f e e t ) , the t e r m i n a l v e l o c i t y , and the drag c o e f f i c i e n t r e s p e c t i v e l y . S^ f o r the body and hind limbs was c a l c u l a t e d by two methods. The f i r s t assumed that each frog's body could be approximated by a s e r i e s of c y l i n d e r s and frustrums of cones, the areas of which could c a l c u l a t e d . The sum of these component areas gave the t o t a l wetted surface area. Secondly, a weight p r o p o r t i o n a l i t y method was used. The dry weight of a standard microscope s l i d e c o v e r s l i p (2.40cm x 3.0cm.) was recorded. I t was then dipped i n t o a viscous soap s o l u t i o n ( S u n l i g h t Dish Detergent). Once removed, the excess soap was allowed to d r i p o f f , l e a v i n g a t h i n l a y e r c o a t i n g the s u r f a c e . The weight of the c o v e r s l i p was recorded again e s t a b l i s h i n g a r e l a t i o n s h i p between u n i t surface area and weight gained. This procedure was followed with the preserved f r o g s . The increase i n weight of the specimens was compared to the standard. The two methods of determining gave values that agreed to w i t h i n 10%. -4 For a l l i n d i v i d u a l s of H. b o e t t g e r i , S w ranged from 3.4x10 to -4 2 5.8x10 m (geometric method). S w f o r the s i n g l e specimen of R. -3 2 p i p i e n s was 7.8x10 m . (geometric method). Patted dry specimens 8 of H. boettgeri and R. pipiens were weighed (Mett l e r M3 microbalance and M e t t l e r PK300 balance r e s p e c t i v e l y ) . Submerged weight was c a l c u l a t e d from: V =W (1-p/p') 5 a (1.2) where W and p' are the weight of the specimen i n a i r , and the a d e n s i t y of the specimen r e s p e c t i v e l y . The submerged weight of each f r o g was changed by i n s e r t i n g lead shot i n t o the body c a v i t y . Experiments were a l s o conducted with i s o l a t e d r e a l and model feet of H. b o e t t g e r i . The planform areas of the feet were estimated by making ink p r i n t s of each foot on paper. The model feet were cut from c e l l u l o s e acetate photocopies of these ink prints-. Real and model feet were mounted on very f i n e i n s e c t pins f o r the drop-tank t e s t s . The pins acted as s t a b i l i z e r s , and were t r e a t e d as point masses with zero drag. Successive cuts along the length of the p i n , a l t e r e d the submerged weight of each f o o t . For H. b o e t t g e r i d r o p - t e s t s were fil m e d at 60 frames sec * (Locam model 51 camera, 4.0-5.6, Kodak Plus Reversal f i l m 7276, ASA 40). L i g h t i n g was provided by two 800 or 1000 watt lamps (Berkey Beam 800 and Berkey C o l o r t r a n Mini King 104-051 r e s p e c t i v e l y ) . Films were d i g i t i z e d with a photographic analyzer (P.A.L. P r o j e c t i o n A n a l y s i s Unit ZAE 76). The t e r m i n a l v e l o c i t y was c a l c u l a t e d from the d i g i t i z e d time -displacement data. Terminal v e l o c i t y was reached w i t h i n 1-2 cm. of r e l e a s e . For R. p i p i e n s , the drop- t e s t s were recorded on video tape with a low l i g h t T.V. camera (R.C.A. TC2011/N). Terminal v e l o c i t y was measured by s i n g l e frame (1/30 sec.) 9 advancement of the recorded displacement data. Specimens f e l l approximately 75cm. p r i o r to recording to ensure t e r m i n a l v e l o c i t y at the time of ta p i n g . Drag force and drag c o e f f i c i e n t s are c a l c u l a t e d from equation 1.1, and are p l o t t e d against t e r m i n a l v e l o c i t y and Reynolds number r e s p e c t i v e l y . Reynolds numbers, R g f o r bodies moving through water can be c a l c u l a t e d from: R e=10.0 6x v e l o c i t y ( m s e c - 1 ) x length(m) (1.3) (Alexander 1971). By s u b t r a c t i n g the d r a g - v e l o c i t y curves f o r spla y e d - l i m b e d / f o o t l e s s specimens, from the curves for complete specimens, the drag and drag c o e f f i c i e n t s of the expanded foot of H. b o e t t g e r i could be c a l c u l a t e d . Flow v i s u a l i z a t i o n experiments were performed with a specimen of H. boettgeri f i t t e d with acetate model f e e t . The t e s t s were conducted i n a p l e x i g l a s s tank, s i m i l a r to that used i n the drop-tank t e s t s . The t e s t specimen was soaked i n a 40% g l y c e r o l s o l u t i o n , h e a v i l y s t a i n e d w i t h f u s c h i n . The frog was then dropped i n t o a tank of c l e a r g l y c e r o l of the same c o n c e n t r a t i o n . The flow p a t t e r n was made v i s i b l e by streams of coloured g l y c e r o l , which were f i l m e d under s i m i l a r c o n d i t i o n s as those of the droptank experiments. 10 RESULTS Drag force increases with increas ing v e l o c i t y and the drag c o e f f i c i e n t decreases with increas ing R g , for v whole-bodied ' foot le s s animals ( F i g . 1 .1-1.3) . For H. b o e t t g e r i , Cp l eve l s off to approximately 0.11 at R g 8000 ( F i g . 1.3) . Cp ranges from 0.06-0.05 at 16600<Re<40400 for R pipiens ( F i g . 1.3) . An ex t rapo la t ion of the C n ~ R e curve for H. boet tger i to the experimental R g range of R. pipiens, and the c D ~ R e curve for R. pipiens are a l so shown in F i g . 1.3. Under dynamical ly s i m i l a r c o n d i t i o n s , the Cp values generated for both species are s i m i l a r , with those of H. boet tger i probably being s l i g h t l y h igher . Drag force increases with increas ing v e l o c i t y for i so l a t ed f rog ' s feet of H. boet tger i ( F i g . 1.4) . Cp values for the i s o l a t e d f rog ' s feet and i s o l a t e d acetate model feet are shown in F i g . 1.5. For i s o l a t e d f rog ' s feet , C.. l e v e l s of f to about 1.6 at R 700. Cn D e D for the model feet range from 1.8-1.2, averaging 1.5, for 300<R <1300. e Figures 1.6-1.9 summarize the Cp r e s u l t s for the foot a lone, based on the subt rac t ion method. F i g . 1.6 shows the d r a g - v e l o c i t y curve for the feet alone (Curve 3) . The corresponding drag c o e f f i c i e n t s are shown i n F i g . 1.7. The subtrac t ion method gives high Cp values for the foot at low R g , compared to those of the i s o l a t e d foot at comparable R^. The two curves approach one another and in te r sec t at R g 1800. The curves give s i m i l a r values of C„ for the foot wi th in a range of 1250<R <2250. The subtrac t ion D e method was a l so appl ied to whole-bodied frogs f i t t e d with model 11 Fig. 1.1 Drag force on the body and hind limbs (feet removed) of H. boettgeri is plotted against velocity for three individuals. Angles of the hind limb segments ranged from 80°-110°/ 130o-180°, and 110°-180°, for the femoral (f), tibiofibular ( t f ) , and metatarsal-phalangeal (mp) segments respectively (see inset). The _ 2 176 curve of best f i t is given by Drag=2.44x10 Velocity ' (n=26, r=0.92, t=11.29, and p<0.001). 12 Fig. 1.2 Drag force on the body and hind limbs (feet removed) of R. pipiens is plotted against velocity for a single individual. The orientation of the body and hind limbs with respect to the incident flow is similar to the inset of Fig. 1.1. The curve of -1 189 best f i t is given by Drag=l.93x10 Velocity " (n=32, r=0.99, t=45.97, and p<0.001). 13 F i g . 1.3 The drag c o e f f i c i e n t s of the body and hind limbs (feet removed) for H. boettgeri ( • ) , and R. pipiens ( • ), are plotted against Reynolds number. The curve of best f i t for H. boettgeri i s given by, C =3.64R u ' (n=26, r=0.75, t=5.49, p<0.001). The ^ -1 -0 108 curve of best f i t for R. pipiens i s given by, CD=1.66x10 R g (n=32, r=0.44, t=2.66, p<0.02). The two curves are compared by the extrapolation of the C n-R g c u r v e f ° r boettgeri ( — —) to the experimental R range of R. pipiens. 14 0.0002n 1 1 1 1 0 0.05 0.1 0.15 VELOCITY M/SEC Fig. 1.4 Drag on the foot of H. boettgeri, oriented normal to the flow (see inset), is plotted against velocity, based on a single specimen. The curve of best f i t is given by, Drag=9.43xlO - 3Velocity 1' 8 1 (n=13, r=0.99, t=52.3, p<0.001). 15 Fig. 1.5 The drag coefficients of the foot of H. boettgeri, isolated frog foot (•), and isolated model acetate foot ( • ), oriented normal to the flow (see inset), are plotted against Reynolds number. The curve of best f i t for the isolated frog foot is given by, C p=6.0R e~°' 1 8 9 (n=13, r=0.85, t=5.08, p<0.001). The average value of Cp for the isolated model acetate foot is 1.50, (n=32, s=0.17, for 300<R <1300), based on four specimens. 16 VELOCITY M/SEC Fig. 1.6 Curve 1 represents the best-fitting power curve for the body, hind limbs, and expanded feet, of H. boettgeri (Drag=2.54xl0" 2Velocity 1 , 3 7, n=22, r=0.99, t=31.55, and p<0.001) based on two individuals. The hind limbs segments were at angles of 80°-90°, 150°-175°, 100°-125°, and 90°, for the femoral, tibiofibular, metatarsal-phalangeal, and foot segments respectively, with respect to the long axis of the body (see inset). Curve 2 represents the curve of best f i t for the body and _2 2_ 76 hind limbs (feet removed), (Drag=2.44x10 Velocity ' , n=26, r=0.92, t=11.29, p<0.001). Curve 3 represents the drag of the feet alone, based on the subtraction method (Curve 3= Curve 1-Curve 2). 17 Fig. 1.7 The drag coefficient of the foot, based on the -2 -0 8 27 subtraction method Curve 1 (CD=G.5xl0 Rg ' ), is compared to Curve 2, the results obtained from the drop-tank tests on the isolated frog foot (from Fig. 1.5). 18 VELOCITY M/SEC Fig. 1.8 Curve 1 represents the best-fitting power curve for the drag of the body, hind limbs, and model feet, of H. boettgeri (Drag=2.65xlO" 2Velocity 1' 6 1, n=9, r=0.99, t=31.55, p<0.001). The curve of best f i t for the drag on the previous individual, with -2 1 62 model feet removed, (Drag=l.48x10 Velocity ' , n=12, r=0.99, t=22.35, p<0.001) is Curve 2. Curve 3 represents the drag of the model foot alone, based on the subtraction method (Curve 3=Curve 1-Curve 2). 19 Fig. 1.9 The drag coefficient of the model foot based on the subtraction method Curve 2 (CD=2.17xlO^Re 0-408^ ^ g c o m p a r e ( 3 to the drag coefficient of the isolated frog foot Curve 1, the isolated model average Curve 3, and the drag coefficient for fl a t circular and square plates oriented normal to the flow Curve 4, taken from Hoerner (1965). 0 represents the drag coefficient of angelfish pectoral fins, calculated by subtraction, from data taken from Blake 1981b. 20 feet ( F i g s . 1.8-1.9). The d r a g - v e l o c i t y curve for the model foot alone, based upon the s u b t r a c t i o n method, i s shown i n F i g . 1.8 (Curve 3). F i g . 1.9 shows the corresponding C n-R g curve for the model foot alone, along w i t h the i s o l a t e d f r o g ' s foot curve, the i s o l a t e d model average, the l i t e r a t u r e value of C D f o r f l a t p l a t e s o r i e n t e d orthogonal to the d i r e c t i o n of flow (Hoerner 1965), and a s u b t r a c t i o n treatment of a n g e l f i s h p e c t o r a l f i n data taken from Blake 1981b. The curve f o r the acetate foot obtained by the s u b t r a c t i o n method i n t e r s e c t s the curve for the i s o l a t e d frog's foot at R of approximately 300. For 600<R <1500, the average model values of C n are i n good agreement with the r e s u l t s obtained by the other two methods. The drag c o e f f i c i e n t f or f l a t p l a t e s normal to the d i r e c t i o n of flow, are s i m i l a r to the 'experimental values obtained f o r the foot for 1000<R <3500. C_ values f o r e D a n g e l f i s h p e c t o r a l f i n are i n good agreement with the frog foot data and the l i t e r a t u r e values taken from Hoerner. Re s u l t s of a t y p i c a l flow v i s u a l i z a t i o n experiment are shown i n F i g . 1.10. As the f r o g moves through the f l u i d , a large r e c i r c u l a t i n g vortex i s generated behind each model f o o t . T y p i c a l l y the vortex forms at the proximal edge of the foot and r o t a t e s d i s t a l l y . The vortex breaks down i n t o l e s s w e l l defined eddies, downstream of the animal. The f l u i d flows r e l a t i v e l y smoothly over the body and hind limbs of the animal. 21 Fig. 1.10 Results of a typical flow visualization experiment with a specimen of H. boettgeri fitted with model acetate feet, are illustrated. Stippling indicates regions of the flow made visible by the coloured glycerol. Regions of vorticity are indicated behind the feet. The specimen f e l l at 0.125msec . The time interval between successive illustrations is 0.5 sec. 22 DISCUSSION Sources of e r r o r must be c o n s i d e r e d . E r r o r i n the c a l c u l a t i o n of the drag c o e f f i c i e n t (CD) can be due to e r r o r i n the submerged weight, the t e r m i n a l v e l o c i t y , and the wetted s u r f a c e area of the specimen (W , U, and S r e s p e c t i v e l y ) . E r r o r i n measuring W i s s w s u n l i k e l y , the balances had r e p r o d u c i b i l i t i e s of 0.1u>g. and 0.50mg. (M e t t l e r M3 microbalance and M e t t l e r PK 300 Balance r e p e c t i v e l y ) . ¥ f o r the s m a l l e s t unloaded specimen was approximately 26.0mg. Terminal v e l o c i t y was measured by d i g i t i z i n g the displacement data of the dropped specimens. The framing r a t e of the c i n e and video cameras f i x e d the time i n t e r v a l . E r r o r i n the c a l c u l a t i o n of U would be due to n o n l i n e a r i t y i n the d i g i t i z e r , human i n a c c u r a c y i n the d i g i t i z i n g p r o c e s s , or a combination t h e r e o f . The p r o j e c t i o n a n a l y s i s u n i t has a l i n e a r i t y of 0.1%, and a r e s o l u t i o n of <0.13mm. I t i s u n l i k e l y t h a t the e s t i m a t i o n of U c o u l d c o n t r i b u t e t o any s e r i o u s e r r o r i n Cp. The two methods of c a l c u l a t i n g the t o t a l wetted s u r f a c e area of the hind limbs and body gave r e s u l t s t h a t were w i t h i n 10%. So, the o v e r a l l e r r o r i n W , U, S and ' s' ' w c onsequently Cp, i s probably s m a l l . D i r e c t measurement of the drag c o e f f i c i e n t of the f o o t of H. b o e t t g e r i (100<Rfi<700, 2.5>CD>1.6, see F i g . 1.5) support the assumption t h a t animal paddles behave s i m i l a r i l y to man-made f l a t p l a t e s o r i e n t e d normal to the flow (100<R e<1000, C D=1.2 in Hoerner 1965). C Q measurements of i s o l a t e d a c e t a t e model f e e t a l s o support t h i s (300<R e<1300, 1.8>C D>1.2 , see F i g . 1.5). Flow pattens seen behind the f e e t ( F i g . 1.10) are c o n s i s t e n t with the high C_ 23 measurements. The drag on whole specimens, with i n t a c t expanded f e e t , appeared to decrease at higher v e l o c i t i e s (see F i g . 1.6). Lack of muscular t e n s i o n i n the preserved specimens r e s u l t e d i n bending of the f e e t , and a decrease i n drag. C n v a l u e s f o r the f r o g ' s f o o t , based on the s u b t r a c t i o n method, were i n c o n s i s t e n t with r e s u l t s obtained with i s o l a t e d f r o g ' s f e e t ( F i g . 1.5). During the power s t r o k e , the f e e t are held r i g i d and expanded by muscular t e n s i o n . R e p l a c i n g the r e a l f e e t with a c e t a t e model f e e t , gave the necessary r i g i d i t y , l i k e l y r e f l e c t i n g the l i v i n g system. The s u b t r a c t i o n method gave Cp v a l u e s f o r the model f e e t t h a t were i n good agreement with the i s o l a t e d model average, the curve f o r the i s o l a t e d f r o g ' s f e e t , and the l i t e r a t u r e value f o r f l a t p l a t e s (Hoerner 1965) ( F i g . 1.9). A s i m i l a r s u b t r a c t i o n method with drag measurements f o r a n g e l f i s h with p e c t o r a l f i n s s e t broadside to the flow, and amputated, (taken from Blake 1981b), gave C n values f o r the p e c t o r a l f i n t h a t were l n good agreement with those obtained f o r the f r o g ' s f o o t . The s u b t r a c t i o n method g i v e s reasonable e s t i m a t e s of the drag c o e f f i c i e n t s of d e l i c a t e s t r u c t u r e s , provided t h a t they remain r i g i d throughout the experiments. The drag c o e f f i c i e n t s f o r H. b o e t t g e r i and R. p i p i e n s can be compared by e x t r a p o l a t i o n . T h i s process c o u l d i n v o l v e extending the Cp-R e curve d e r i v e d f o r H. b o e t t g e r i , to the experimental R g range of R. p i p i e n s , or v i c e - v e r s a . The change i n Cp f o r R. p i p i e n s over the experimental Rfi range was s m a l l . In c o n t r a s t , Cp changed s u b s t a n t i a l l y f o r H. b o e t t g e r i , beginning to l e v e l out at the upper end of the experimental R g range. Because the experiments done with H. b o e t t g e r i occured over the range of 24 greatest change for the Cp-R g curve , an extension of the curve der ived for ti. boet tger i was s e l e c t e d . F i g . 1.3 shows that the C D values for these two species are s i m i l a r , over a comparable Reynolds number range. R. p ip i ens i s u s u a l l y found i n meadows, often d i s t a n t from open water. I t overwinters i n s t i l l or f lowing water, moving to smal ler pools i n the sp r ing to breed (Porter 1967, Dickerson 1969 and Schueler 1982). In c o n t r a s t , ti. boe t tger i i s almost whol ly aqua t i c , employing swimming as i t s p r i n c i p a l means of locomotion (Noble 1931, Von F i l e k 1973). Whiting (1961), showed that the p e l v i c g i r d l e of r an id frogs d i f f e r s s u b s t a n t i a l l y from that of p i p i d s . In r a n i d s , the p e l v i c g i r d l e al lows v e r t i c a l bending, and t h i s i s thought to be important i n l and ing . In p ip ids : , which are l a r g e l y burrowers and swimmers, the p e l v i c g i r d l e al lows l o n g i t u d i n a l s l i d i n g , e f f e c t i v e l y lengthening the cyc le of l imb movement. A comparison of the musculature of ranids (R. catesbeiana ( G i l b e r t 1976)), and ti. boettgezl a l so shows Important d i f f e r e n c e s . In ti. b o e t t g e r i , abdominal muscles run p a r a l l e l to the backbone, and extend to the femur (a lso shown i n P i p a , Whiting 1961). In r a n i d s , the abdominal muscles l a t t e r a l l y circumvent the abdomen, and there i s no attachment to the femur. Despite s u b s t a n t i a l i n t e r n a l s t r u c t u r a l d i f f e rences R. p ip iens and ti. boe t tger i are charac ter i zed by s i m i l a r Cp values at comparable R f i. Based on the Cp r e s u l t s , i t seems that adaptat ion for good jumping performance i n R. p ip i ens does not compromise i t s swimming a b i l i t y . I t may be that ti. boe t tger i and R. p ip iens are not s u f f i c i e n t l y d i s t a n t , i n terms of locomotor or e c o l o g i c a l 25 s p e c i f i c i t y , to warrant notable d i f f e r e n c e s i n C n. Further comparative s t u d i e s of f u n c t i o n a l morphology i n r e l a t i o n to locomotion are needed to f i r m l y e s t a b l i s h whether or not design compromises e x i s t f or jumping and swimming performance i n anurans. 26 CHAPTER I I MOTION ANALYSIS AND ESTIMATION OF THE PROPULSIVE FORCE GENERATED BY THE FROG HYMENOCHIRUS BOETTGERI ABSTRACT Cine f i l m s were used to study swimming i n the f r o g , Hymenochirus b o e t t g e r i d u r i n g n e a r - v e r t i c a l b r e a t h i n g e x c u r s i o n s . The animals g e n e r a l l y d e c e l e r a t e d d u r i n g hind limb f l e x i o n ( r e c o v e r y phase), and a c c e l e r a t e d throughout hind limb e x t e n s i o n (power phase). Body v e l o c i t y p a t t e r n s of fro g s are d i s t i n c t from those of other drag-based p a d d l e r s , such as a n g e l f i s h and water boatman, where the body i s a c c e l e r a t e d and d e c e l e r a t e d w i t h i n the power s t r o k e phase. The p r o p u l s i v e f o r c e generated d u r i n g hind limb e x t e n s i o n was estimated f o r a s i n g l e sequence, from q u a s i - s t e a d y drag and i n e r t i a l c o n s i d e r a t i o n s . The upper and lower bounds of t h i s estimate were c a l c u l a t e d by c o n s i d e r i n g a d d i t i o n a l components of the f o r c e balance, i n c l u d i n g the net e f f e c t of g r a v i t y and buoyancy, and the l o n g i t u d i n a l added mass f o r c e s a s s o c i a t e d with the f r o g ' s body. The v a l i d i t y of using s t a t i c drag estimates to d e s c r i b e dynamic r e s i s t a n c e i s d i s c u s s e d . S t a t i c drag estimates probably approach dynamic v a l u e s , when l a r g e , f a i r l y r i g i d bodies are p r o p e l l e d by r e l a t i v e l y s m a l l appendages. The s t a t i c drag c o e f f i c i e n t ( C p - s t a t i c , from Chapter I) approaches the drag c o e f f i c i e n t based on the d e c e l e r a t i o n of the body d u r i n g hind limb f l e x i o n (C_-dynamic), d u r i n g the l a t t e r stages of f l e x i o n when the hind limbs are sp layed . C p - s t a t i c i s a poorer estimate of Cp-dynamic when the animal i s s t reaml ined . This quasi-steady approach to quant i fy ing the dynamic drag, may be useful i f a range of C p - s t a t i c va lues , corresponding to the hind limb o r i en ta t ions throughout the c y c l e , were used. The estimate of the propul s ive force remains p o s i t i v e throughout extens ion, d i s t i n c t from the blade-element force c a l c u l a t i o n s of drag-based paddlers ( ange l f i sh and waterboatman). It i s suggested that simple drag-based propul s ion may not be s u f f i c i e n t to exp la in the swimming patterns observed i n f rogs . 28 INTRODUCTION Est imat ing the propuls ive hydrodynamic forces generated by the locomotor apparatus of a swimming animal can be s i m p l i f i e d by cons ider ing the force balance for swimming at a constant v e l o c i t y and l e v e l i n the water column: T=D and L=W where T , D, L , and W, represent t h r u s t , drag , l i f t and weight r e s p e c t i v e l y . Studies i n v o l v i n g thrust product ion i n n e u t r a l l y buoyant animals e l iminate the neces s i ty to consider l i f t forces dur ing constant l e v e l swimming. When constant forward v e l o c i t y i s measured, a further s i m p l i f i c a t i o n can be made. In the non-acce lera t ing system, thrust i s equal to drag. The force balance has been used to study the energet ics of drag-based paddl ing and rowing in some aquat ic and semi-aquatic animals (Prange and Schmidt-Neilsen i n duck 1970, DIPrampero et a l i n humans 1974, Prange i n sea t u r t l e 1976, Kemper et a l . i n humans 1983, and Wil l iams i n mink 1983). Blake (1979) inves t iga ted drag-based propul s ion i n the a n g e l f i s h , and using blade-element theory, developed a general model for es t imat ing the fo rce , power, and e f f i c i e n c y of drag-based paddlers . This model has been further used to examine swimming i n muskrat (F i sh 1984). Blake develops the theory further and app l ie s i t to swimming in the water boatman (Blake 1986). Some anecdotal references have been made to swimming i n frogs i (eg. Porter 1967, and Beebee 1985), yet no d e t a i l e d study has been 29 done. Calow and Alexander (1973) inves t iga ted the mechanics of jumping i n Rana tempozazia using a synchronized force p la te and c ine f i l m technique. They ca l cu l a ted the force generated by the hind limbs during l i f t - o f f . For comparison, they estimated the force generated dur ing hind limb extension i n swimming, by cons ider ing the a c c e l e r a t i v e and dece le ra t ive phases of a v e l o c i t y - t i m e curve der ived from cine data . The force developed by the hind limbs during the jump take-of f was found to be three times greater than that estimated for swimming (both maximal e f f o r t s ) . Here, v e l o c i t y and a c c e l e r a t i o n records of swimming H. boet tger i are made from high-speed c ine f i l m s . The v e l o c i t y p r o f i l e s are compared to those of other rowers. The average forward force generated dur ing extension of the hind limbs i s estimated from the force balance. The c a l c u l a t i o n uses the s t a t i c - b o d y drag c o e f f i c i e n t (Chapter I) and instantaneous v e l o c i t y measurements, to e s t a b l i s h the quasi-steady drag experienced by the animal . The a c c e l e r a t i o n records give information about the net i n e r t i a l forces a c t ing on the animal . By a d d i t i o n , the t o t a l forward force can be est imated. The d i s c u s s i o n focuses on the assumptions associated with t h i s e s t imat ion , p a r t i c u l a r i l y the v a l i d i t y of using s t a t i c drag measurements to represent the re s i s tance of dynamic systems. A comparison of t h i s es t imate , to force c a l c u l a t i o n s based on blade-element model l ing i n other paddler s , w i l l a l so be made. Frogs have been catagor ized as drag-based rowers (Blake 1981, Webb and Blake 1982), mainly because of t h e i r large webbed feet . However, un l ike other rowers that have been s tudied (eg. ange l f i sh 30 and waterboatman), the caudal placement of t h e i r p r o p u l s i v e legs i n d i c a t e s the p o t e n t i a l f o r i n t e r a c t i v e e f f e c t s , which are not acknowledged i n the blade-element model. Comparing the whole-body kinematics of H. boettgeri with other paddling animals, may i n d i c a t e the r e l a t i v e importance of hind limb i n t e r a c t i o n . 31 MATERIALS AND METHODS Healthy animals were maintained on a d i e t of br ine shrimp, i n a f i v e g a l l o n l abora tory aquarium, equipped with a r e c i r c u l a t i n g f i l t e r . The water temperature var ied with the ambient room temperature ( 2 0 ° - 2 5 ° C ) . Frogs were f i lmed at 500 frames s e c - 1 (Locam model 51 c ine camera) dur ing n e a r - v e r t i c a l breathing excurs ions . Exposure was adjusted for the a c c e l e r a t i o n of the camera (see F i g . 2 .1 ) . An opaque p l e x i g l a s s p a r t i t i o n , marked with a 0.50x0.50cm. g r i d , was placed into the tank, approximately 5.0cm. from and p a r a l l e l to the front g lass pannel . The camera was f ixed to the f loor about 1.5-2.0m. from the front of the tank,, and flanked by two 800 or 1000 watt f lood lamps (Berkey Beam 800 and Berkey C o l o r t r a n Mini King 104-051 r e s p e c t i v e l y ) . The g r i d was brought in to sharp focus . The f i e l d of view was marked on the f ront g lass pannel , then bounded by two more p l e x i g l a s s p a r t i t i o n s . E ight to ten animals were placed in to t h i s enclosed area . The set-up i s shown i n F i g . 2 .2 . When any animal swam through the f i e l d of view, the lamps and camera were ac t iva ted s imultaneously by a s i n g l e power bar . F i l m (8 x 100 f t . , ASA 400 Kodak 4X) was shot , processed, and inspected for good sequences. A good sequence was judged by the maintainance of a s t r a i g h t path i n a plane p a r a l l e l to the front g lass pannel and g r i d , and by the symmetry of the limb s t roke . The se lec ted sequences were analyzed frame by frame with a photograghic a n a l y s i s un i t ( P . A . L . photographic a n a l y z e r ) . The 32 0 100 200 300 400 NUMBER OF FRAMES Fig. 2.1 The Locam speed curve (for model: 50-0003 and 51-0003, 200'-400' LOCAM AC, 100' acetate film, page 30 of the Locam Instruction Manual, REDLAKE Corporation 1979) was transformed from frames second as a function of film length, to frames second * as a function of number of frames (40 frames foot * ) . Data taken from the original speed curve was f i t to a cubic polynomial (y=1.6+3.7X-1.2xl0~2X2+1.6xl0"5X3, r=0.999, F7703.8). The camera reached 500 frames second * in about 330 frames. The number of frames were counted from camera activation, to the f i r s t frame of a viable sequence. Where sequences, or portions thereof, were contained within the f i r s t 330 frames, the speed curve equation was used to calculate the instantaneous frame rate and exposure. 33 Fig. 2.2 Front (a) and side (b) views of the cine film set-up are shown. The camera, flanked by two 800 or 1000 watt lamps, is fixed to the floor about 1.5-2.0m. from the front glass pannel. The distance between the front glass pannel and the vertical grid is about 5cm. Further details are given in the MATERIALS AND METHODS. 34 t i p of the snout and vent were d i g i t i z e d , a l lowing for the c a l c u l a t i o n of instantaneous snout-vent l ength . Four sequences were chosen for the a n a l y s i s . The cummulative displacement of the t i p of the vent was p lo t t ed as a funct ion of t ime. A seven-point moving polynomial regres s ion was employed to smooth the data , as fo l lows . The f i r s t seven data points (displacement, time) were f i t t e d to a quadrat ic f u n c t i o n . A s i n g l e new displacement value was generated by s o l v i n g t h i s funct ion at the fourth time increment i n the seven point s e t . The f i r s t and second d e r i v a t i v e s of t h i s funct ion were then determined. Eva luat ing these new functions at the same time increment, gave the instantaneous v e l o c i t y and a c c e l e r a t i o n of the vent of the animal , corresponding to the b e s t - f i t displacement. The data set was moved by p i c k i n g up the next sequent ia l point and dropping the f i r s t one of the o r i g i n a l data se t , thereby maintaining seven p o i n t s . The above process was repeated throughout the e n t i r e data se t , to give smooth displacement, instantaneous v e l o c i t y , and acce lera t ion- t ime records for each animal . The average forward force produced by a swimming animal was est imated, us ing the v e l o c i t y and a c c e l e r a t i o n records from sequence 1, as f o l lows . A non-zero a c c e l e r a t i o n r e f l e c t s the presence of a net force ( F n e i . ) ' F n e t * s t n e d i f f e rence between T (the t o t a l forward force or t h r u s t ) , and D (drag, the r e s i s t i v e f o r c e ) , F =T-D net (2.1) 35 Therefore , T = F n e t + D (2.2) Drag i s a funct ion of the square of v e l o c i t y , D = l / 2 p S C . U 2 (2.3) V D where p, S y , C n , and U are f l u i d d e n s i t y , t o t a l wetted surface area , drag c o e f f i c i e n t and v e l o c i t y r e s p e c t i v e l y . The drag c o e f f i c i e n t of the body and splayed hind limbs of H. boet tger i was c a l c u l a t e d as a funct ion of Reynolds Number: C n = 3 . 6 4 R e " 0 , 3 7 8 (2.4) (Chapter I , F i g . 1 . 3 ) Using the snout-vent length corresponding to the splayed-1imbed o r i e n t a t i o n , the instantaneous R f i was c a l c u l a t e d from: R f i=106x v e l o c i t y f m s e c " 1 ) x length(m) (2.5) (Alexander 1971). The t o t a l vet ted surface area of the animal ( S y 2 i n metres ) was estimated by a s c a l i n g r e l a t i o n s h i p based on surface area measurements of preserved frogs (geometric surface area determinat ion , Chapter I ) : S = 0 . 1 8 8 \ 1 , 5 2 (2.6) w 36 where \ i s snout-vent length i n metres (see F i g . 2.3 for log t rans format ion) . The instantaneous (quasi-steady) drag force on the swimming frog was computed and p lo t ted as a funct ion of time for the e n t i r e f l ex ion-ex tens ion c y c l e . F n e t i s a funct ion of the mass and a c c e l e r a t i o n of the an imal : F n e t =ma (2.7) The mass of the frog was estimated by a s c a l i n g r e l a t i o n s h i p der ived from measurements of preserved specimens: mass(kg)=1.031xl0 2 X 3' 1 9 (2.8) (see F i g . 2.4 for log t rans format ion) . F n e t - / computed for the e n t i r e sequence, was p l o t t e d on the same sca le as the drag force (D). D and F n e f . were summed over the per iod of extension to give T . The impulse of T d i v i d e d by the durat ion of the extension ( t p ) , g ives T = l / t p y T d t (2.9) the average forward force (T) generated by the an imal , g iven the locomotor pat tern observed i n sequence 1. Balance readings (Mett ler PK300 balance) were taken of l i v e anaesthet ized (MS222) animals i n a i r and i n water (with the 37 Fig. 2.3 Log of total wetted surface area (S w, m ) is plotted as a function of log snout-vent length ( X , m) for preserved tf. boettgeri ( • ). The curve of best f i t ( ) is logSw=-0.73+1.521og\ (n=6, r=0.925, r s i g=0.917 at p<0.01). The 95% confidence intervals of the predicted S w are Y . ± t 05(2)(n-2 ) S Y. ' W N E R E S Y . i s t n e standard error of the population. S y was measured by the geometric method described in Chapter I, MATERIALS AND METHODS. 38 Fig. 2 .4 Log mass (kg) is plotted as a function of log snout-vent length ( X , m) for preserved H. boettgeri ( • ). The curve of best f i t ( ) is logmass=2.01+3.21ogX (n=9, r=0.957, r s . g = 0.898, at p< 0 . 0 0 1 ) . The 95% confidence intervals of the predicted mass are Y.± t rj5(2) (n-2)**Y.' w n e r e ^y. * s *"^e stan°"ard error of the population. Mass was measured with a Mettler Microbalance. 39 accessory for dens i ty determinat ion) . The change i n the reading ind ica ted the e f fec t of a buoyant force on the animal . Negative buoyancy was ind ica ted by a percent reduct ion i n the balance read ing . Animals that ne i ther f loated nor gave any reading on the balance, ind ica ted neut ra l buoyancy. Those that f loated to the surface were p o s i t i v e l y buoyant. P o s i t i v e buoyancy was not q u a n t i f i e d . An estimate of the added mass of the body was based on the f ineness r a t i o (defined as snout-vent length/average of major and minor axes of the e l l i p t i c a l cross sec t ion at the shou lder ) , and the r e l a t i o n s h i p presented by Landweber (1961). L inear measurements were made of nine preserved specimens (preservat ion as i n Chapter I , MATERIALS AND METHODS) with v e r n i e r c a l i p e r s (Mitutoyo) , and f ineness r a t i o s were c a l c u l a t e d . The f ineness r a t i o of the animal i n Sequence 1 was taken as the average of the nine values (f ineness ra t io=5 .5 , s=0.50, and n=9). 40 RESULTS Two complete f l ex ion-ex tens ion and two extension sequences were chosen for the motion a n a l y s i s . The r e s u l t s of the motion a n a l y s i s and smoothing technique for Sequence 1, a complete hind limb f l e x i o n and extension are shown i n F i g . 2 .5 . The smooth displacement p a r a l l e l s the experimental values ( F i g . 2 .5a) . In F i g . 2 .5b, the experimental Instantaneous v e l o c i t i e s (the numerical d i f f e r e n t i a t i o n of the experimental displacement curve 2.5a) and the instantaneous v e l o c i t i e s generated from the smoothing technique are shown. The smooth values c l a r i f y the t r e n d . F i g . 2.5c shows the smoothed a c c e l e r a t i o n dur ing f l e x i o n and extens ion , with the a c c e l e r a t i o n values generated from the numerical d i f f e r e n t i a t i o n of the experimental instantaneous v e l o c i t i e s from F i g . 2.5b. The smooth record c l e a r l y shows a marked p o s i t i v e a c c e l e r a t i o n , reaching a maximum value about the midpoint of hind limb extens ion . No c l e a r trend i s v i s i b l e i n a c c e l e r a t i o n s der ived from double d i f f e r e n t i a t i o n of the o r i g i n a l displacement da ta . The experimental and smooth instantaneous snout-vent length for Sequence 1 i s shown In F i g . 2 .5d. Snout-vent length decreases and increases approximately 20% throughout f l e x i o n and extension r e s p e c t i v e l y . The displacements , v e l o c i t i e s , and a c c e l e r a t i o n s are based on the progress ion of the vent . Because numerical d i f f e r e n t i a t i o n ampl i f i e s no i s e , i t i s necessary to process displacement-time data to obta in meaningful v e l o c i t y and a c c e l e r a t i o n records . The s e l e c t i o n of a p a r t i c u l a r z LU LU _ : O S < u _ l D_ LO LO LU o U j ' d ZJ LU UJ °1 3 ° o z LU U J O > .1 o z CO OD 0.10 020 TIME SEC -FLEX.4EXT.-I 030 41 Fig. 2.5 The results of the motion analysis and smoothing technique are shown for sequence 1. Experimental (and numerically differentiated) and smooth data are represented by open ( O ) and closed ( • ) circles, respectively. 42 technique depends on the system being s tud ied . The displacement-time data sets for the four sequences were modelled as h i g h l y s i g n i f i c a n t t h i r d , f our th , and f i f t h order polynomials . However, double d i f f e r e n t i a t i o n of each of these funct ions gave very d i f f e r e n t a c c e l e r a t i o n records ( l i n e a r , quadrat ic and cubic r e s p e c t i v e l y ) , which could not be r e c o n c i l e d with the movement of the body and hind l imbs . Therefore , s i n g l e c u r v e - f i t t i n g was re jec ted for t h i s s tudy. The seven-point moving polynomial regres s ion technique was se lec ted ins tead , because i t d i d not force the data to f i t a s i n g l e predetermined p a t t e r n . F i g . 2.6 shows the smooth instantaneous v e l o c i t y records for the four se lec ted sequences, based on the progress ion of the vent . The v e l o c i t y of the body increases throughout hind limb extens ion, to maximum values of about 10-20cmsec~*. During hind l imb f l e x i o n , body v e l o c i t y may decrease s t e a d i l y or remain f a i r l y constant ( F i g . 2.6a and b r e s p e c t i v e l y ) . In F i g . 2 .7 , the a c c e l e r a t i o n records for the four sequences (based on the progress ion of the vent) are shown. The animals acce lera te throughout hind l imb extension ( F i g . 2.7a, b , and d ) . During the l a t t e r part of extens ion , F i g . 2.7c shows a s l i g h t negative a c c e l e r a t i o n . The body may experience negative a c c e l e r a t i o n dur ing hind l imb f l e x i o n ( F i g . 2 .7a) . The cons iderable f luc tua t ions i n the a c c e l e r a t i o n of the body during hind limb f l e x i o n shown i n F i g . 2.7b, ampl i fy the noise of the corresponding v e l o c i t y record ( F i g . 2 .7b) . C p - s t a t i c , based on the vent v e l o c i t y dur ing the hind limb f l e x i o n of Sequence 1 ( F i g . 2.6a) and equation 1.4 (Chapter I ) , i s 43 CH 1 1 1 " « 1 0.15 0.10 0.05 0.0 0.05 0.10 0.15 0.20 | FLEXION 1 EXTENSION 1 TIME SEC. Fig. 2.6 Smooth 'vent' velocity, synchronized at hind limb extension, is plotted as a function of time for each of the four sequences chosen for the analysis. 44 Fig. 2.7 Smooth 'vent' acceleration, synchronized at hind limb extension, is plotted as a function of time for each of the four sequences chosen for the analysis. 45 shown i n F i g . 2 .8 . The mean force associated with the dece le ra t ion of the body during the hind limb f l e x i o n of Sequence 1, may be estimated from the a c c e l e r a t i o n values ( F i g . 2.7a) and the body mass (equation 2 .8 ) . Assuming that t h i s force i s the r e s u l t of s t a t i c body and hind limb drag , al lows for the c a l c u l a t i o n of Cp-dynamic by equation 2.3 (the drag equat ion, where S v i s from equation 2 .G) . The upper and lower bounds of Cp-dynamic were estimated by the maximum mass, minimum S y and minimum mass, maximum S^ values r e s p e c t i v e l y (see F i g . 2.3 and 2 .4 ) . Vent v e l o c i t i e s and acce le ra t ions ( F i g . 2.6a and 2.7a r e s p e c t i v e l y ) were used for a l l Cp c a l c u l a t i o n s i n t h i s f i g u r e . Cp- s ta t ic appears to be a poorer estimate of the drag c o e f f i c i e n t of the body and hind limbs during the i n i t i a l phase of hind limb f l e x i o n , when the animal i s s t reaml ined. During the l a t t e r stages of f l e x i o n , when the hind limbs are sp layed, Cp- s ta t ic Is a good estimate of the drag c o e f f i c i e n t of the body and hind l imbs . 46 Q80i T I M E SEC. F L E X I O N Fig. 2.8 Cp-static ( 1 , calculated from equation 2.4 and the vent velocity) and C^-dynamic ( 2 , calculated from the mean decelerative force of sequence 1 (vent acceleration, mean mass)), are plotted as functions of time during the hind limb flexion of sequence 1. The upper and lower bounds of Cp-dynamic were estimated by maximum mass, minimum S y and minimum mass, maximum S w values respectively. Vent velocities and accelerations were used for a l l Cp-dynamic calculations. The stick figures indicate the orientation of the hind limbs at the i n i t i a l and final stages of flexion. See RESULTS and DISCUSSION for further explanation. 47 DISCUSSION Motion ana lys i s provides information about the net forces operat ing in swimming systems. Body v e l o c i t y of H. boet tger i increased throughout hind limb extension ( F i g . 2.6 a, b, and d ) , decreas ing only s l i g h t l y in the l a t t e r stages of extension i n one sequence ( F i g . 2 . 6c ) . Thi s pat tern i s cons i s tant with the v e l o c i t y record obtained by Calow and Alexander ( F i g . 12, Calow and Alexander, 1973) for a swimming cyc le of the s e m i - t e r r e s t r i a l Rana temporaria . It i s markedly d i f f e r e n t from the v e l o c i t y records of other drag-based propul sor s . V e l o c i t y patterns (with a r b i t r a r y axes) are shown for the power (extension) and recovery ( f lex ion) phases of swimming frogs , a n g e l f i s h , and water boatman (Blake 1979 and 1986 r e s p e c t i v e l y ) in F i g . 2 .9 . The ange l f i sh and water boatman acce lera te and decelerate wi th in the power phase, i n patterns that are s i ze (R g ) dependent. The ange l f i sh (8.0cm.) maintains approximately constant body v e l o c i t y throughout the c y c l e . The water boatman (0.85cm.) stops at the end of the power phase, and moves s l i g h t l y backward during recovery . This pat tern i s more exaggerated i n the n a u p l i i larvae (0.08cm) of c e r t a i n crustaceans (R.W. B lake , personal communication). One might p r e d i c t H. boet tger i (snout-vent length 2* 2.0 cm.) to show an o s c i l l a t o r y pat tern between the extremes of the ange l f i sh and water boatman, on the bas is of s i z e . Thi s i s not the case. Moreover, H. boet tger i and R. temporaria show a s i m i l a r a c c e l e r a t i v e pat tern throughout the power and recovery stroke despi te a twenty f o l d mass d i f f e rence (== l . O g . and ^ 20.Og 4 8 Fig. 2.9 The body velocity patterns of the angelfish ( 1 ), water boatman ( 2 ), and the frog ( 3 , H. boettgeri and R. temporaria (from Calow and Alexander 1973) are illustrated during the power and recovery phases of a single swimming cycle. Velocity maxima and minima are arbitrary. An alternate pattern suggested for H. boettgeri ( 4 ) , is based on size. See DISCUSSION for further explanation. 49 r e s p e c t i v e l y ) . These d i f fe rences suggest that simple drag-based propul s ion may not be s u f f i c i e n t to descr ibe the locomotor behavior of f rogs . Es t imat ing T and T The propuls ive force (T) generated dur ing the hind limb extension of H. boet tger i i s shown i n F i g . 2.10. The upper and lower bounds of T and T were est imated, tak ing account of the change i n shout-vent length during hind limb f l e x i o n and extens ion , the net e f fec t of buoyancy and g r a v i t y during breathing excurs ions , and the l o n g i t u d i n a l added mass forces associated with the body (C D based on the s t a t i c splayed-l imb o r i e n t a t i o n (Chapter I ) . Because the snout-vent length increases and decreases with the extension and f l e x i o n of the hind limbs r e s p e c t i v e l y (see F i g . 2 .5d) , the instantaneous v e l o c i t y and a c c e l e r a t i o n of the vent (see F i g . 2.6a and 2.7a r e s p e c t i v e l y ) underestimate the corresponding values for the snout. When t h i s i s co r rec ted , upper (snout) and lower (vent) estimates of the instantaneous v e l o c i t y and a c c e l e r a t i o n of the body can be made. Observations of these nega t ive ly buoyant animals dur ing r e s p i r a t o r y excursions i n the l abora tory hold ing tank, show that they u s u a l l y make n e a r - v e r t i c a l ascensions , often r e l e a s i n g an a i r bubble immediately before breaking the sur face . A i r Is q u i c k l y gulped, and they re turn to the subs tra te , along v i r t u a l l y the same path . Buoyancy r e g u l a t i o n occurs at the tank f l o o r , where i f 50 • o 2.5 i 2.0-1.5 1.0 LU O rr S Q5 0.0 -0.5 T=17O10 -3 002 0D6 0.10 TIME SEC. 0.U 0I8 — E X T E N S I O N -Fig. 2.10 The quasi-steady drag ( 1 ) and inertial forces ( 2 ) associated with sequence 1 are plotted as functions of time. The predicted total thrust, T (m—m) is shown, with upper and lower bounds (stippling). The upper limit of T (upper border of stippling) and T is given by the sum of : maximum quasi-steady drag D, maximum F fc, a longitudnal added mass coefficient (of the body) of 0.10, and a net buoyant force of 7.0% of the animal's maximum weight (from the upper bound of Fig. 2.4). The lower limit of T (lower border of stippling) and T is given by the sum of the minimum quasi-steady drag D, minimum F n e t / and a net gravitational force of 7.0% of the animal's maximum weight. This lower limit is futher decreased (----) i f the Cp-dynamic value corresponding to the streamlined orientation (from Fig. 2.8) is used to calculate the minimum quasi-steady drag, D. See DISCUSSION for further explanation. 51 necessary, a i r bubbles are released u n t i l the animals remain s t a t i o n a r y . S imi l a r regula tory behavior has been reported i n Xenopus laevis (another p i p i d ) , where i t i s thought to minimize v u l n e r a b i l i t y to a e r i a l predat ion (Baird 1983). During descent therefore (as i n sequence 1) , H. boet tger i may be p o s i t i v e l y , n e u t r a l l y , or nega t ive ly buoyant. A p o s i t i v e buoyant force would tend to re tard t h e i r e f f o r t s i n a d d i t i o n to drag. Neutra l buoyancy would have no e f f ec t on the force balance. Negative buoyancy impl ies a downward force component. Determining the buoyant s tate of the l i v e swimming animals i s d i f f i c u l t . A i r and submerged weights were determined for l i v e anaesthetized specimens. Some animals were p o s i t i v e l y buoyant. Others were n e u t r a l l y buoyant. The submerged weights of nega t ive ly buoyant specimens ranged from 1.5-7.0% of t h e i r a i r weights. The upper and lower bounds of T and T are inf luenced by p o s i t i v e and negative buoyant force of 7% of the animal ' s a i r weight, r e s p e c t i v e l y . When a body i s acce lera ted through f l u i d , a mass of f l u i d i s acce lera ted with i t . Thi s added mass i s dependent upon s i z e , volume, shape, type of motion, and the f l u i d dens i ty (Batchelor 1967 p. 407). It e f f e c t i v e l y increases the mass of the system. Technica l equiva lents are used to determine the added mass c o e f f i c i e n t s of organisms, analagously to the assignment of drag c o e f f i c i e n t s . Because of the i r r e g u l a r shape of the f rog ' s body, t e c h n i c a l equiva lents are l a c k i n g . The added mass c o e f f i c i e n t was based on the f ineness r a t i o of the torso of the animal (5 .5 , n=9, s=0.50 see MATERIALS AND METHODS) and the r e l a t i o n s h i p 52 presented by Landweber (1961). The upper bound of T and T i s based on an added mass c o e f f i c i e n t of 0.10. Added mass forces are considered n e g l i g i b l e and neglected in es t imat ing the lower bound of T and T. S t a t i c drag measurements may not r e f l e c t the res i s tance experienced by a swimming body. The r e l a t i v e r i g i d i t y of the body probably plays an important r o l e i n determining when s t a t i c estimates approach dynamic r e a l i t i e s . L I g h t h i l l (1971), Inferred that the drag on an undulat ing body, c h a r a c t e r i s t i c of f i s h that swim i n the carangiform and sub-carangiform mode, may be up to f ive times greater than that on a s i m i l a r r i g i d streamlined body. This drag augmentation was thought to be the r e s u l t of boundary layer compression and subsequent increases i n shear s tresses at the f lu id-body in ter face (Bone i n L i g h t h i l l 1971). Prange (1976) j u s t i f i e d h i s use of s t a t i c drag measurements with sea t u r t l e s because of t h e i r r i g i d s h e l l , arguing that the hydrodynamic complexi t ies inherent i n such measurements on f i s h or aquat ic mammals which change t h e i r shape as they swim, do not e x i s t i n the sea t u r t l e . F i s h (1984) based h i s muskrat swimming energet ics s tudies on dead drag measurements with frozen animals , because he found no appreciable f l e x i o n of the body during swimming. Many r i g i d - b o d i e d animals , however, are prope l l ed by o s c i l l a t i n g appendages. Thei r r e l a t i v e s i ze and o r i e n t a t i o n may s i g n i f i c a n t l y a l t e r the flow about the body, from that of the s t a t i c equ iva l en t . N a c h t i g a l l (1977), commented that the moving legs of the water beet le would s l i g h t l y modify the s t reaml in ing c h a r a c t e r i s t i c s of the t runk, and the pressence of the trunk would 53 cause c e r t a i n dev ia t ions from the free water operat ion of the rowing apparatus, but contended that the i n t e r a c t i o n s would be n e g l i g i b l e . Fur ther , he stated that unl ike undulat ing swimmers whose drag/ thrust generating s t ructures are integrated and are not e a s i l y d i s t i n g u i s h e d , waterbeetles (Dytlsc idae) represent swimming systems whose hydrodynamically important s t ructures ( the i r swimming legs) s c a r c e l y i n t e r a c t . Given the d e l i c a t e s t ructure and l a t t e r o - v e n t r a l placement of the swimming legs , N a c h t i g a l l ' s comment seems reasonable. However, the unsteady movement of even small appendages may not be t r i v i a l . Blake (1986), showed that the acce lera t ive-based thrust produced by the swimming legs of the water boatman (Cenocorixa b i f i d a ) represented a s i g n i f i c a n t por t ion (about 33 %) of the t o t a l propuls ive force, generated. Inferred Cp values for the body (based on the force balance) agreed with experimental ly-determined drag c o e f f i c i e n t s for insec t s operat ing at s i m i l a r Reynolds numbers. DiPrampero et a l . (1974, f ront crawl i n humans) commented that although body drag has genera l ly been assumed to be equal to that measured on p a s s i v e l y towed sub jec t s , a swimmer In motion presumably experiences a higher drag, due to movements of the head, l imbs , and t runk, more complex wave formation, and changes i n buoyancy due to r e s p i r a t i o n . Body and/or limb o s c i l l a t i o n s tend to augment drag . R i g i d body estimates of drag probably approach r e a l dynamic values when large r i g i d bodies are prope l led by r e l a t i v e l y s m a l l , s teadi ly-moving appendages. The movement of the f rog ' s body; however, i s much d i f f e r e n t than the s i t u a t i o n s discussed p r e v i o u s l y . During hind limb 54 extens ion, the frog becomes i n c r e a s i n g l y s t reaml ined . I ts p r o f i l e area decreases . It could be argued that the t o t a l wetted surface area (Sw) a l so decreases during extens ion, as the inner surfaces of the hind limbs become shie lded from the f l u i d . Despite an increase in v e l o c i t y , drag i s l i k e l y to decrease with increas ing extens ion . F i g . 2.8 i l l u s t r a t e s t h i s p o i n t . Drag c o e f f i c i e n t s determined from the instantaneous dece le ra t ion of the body during hind limb f l e x i o n approach s t a t i c C n va lues , when the hind limbs are sp layed , as those specimens used to make the i n i t i a l s t a t i c drag measurements (Chapter I ) . C p - s t a t i c Is a poor r e f l e c t i o n of dynamic drag when the animal i s s t reaml ined , i n d i c a t i n g the shape dependence of the drag c o e f f i c i e n t . This quasi-steady approach to quant i fy ing the dynamic drag, may be useful i f a range of C p - s t a t i c va lues , corresponding to the hind limb or i en ta t ions throughout f l e x i o n and extens ion, i s used. Sources of e r ror i n the ac tua l c a l c u l a t i o n of T and T should be cons idered . F n e ^ a n < ^ * n e quasi-steady drag components required the es t imat ion of the mass and t o t a l wetted surface area (S ) w r e s p e c t i v e l y . These estimates were based on s c a l i n g r e l a t i o n s h i p s with snout-vent length (see MATERIALS AND METHODS). Unfor tunate ly , the data bases for these s c a l i n g r e l a t i o n s h i p s were small (n=6 and 9 for S^ and mass r e s p e c t i v e l y ) . The animal corresponding to sequence 1 was s l i g h t l y smaller that the smal lest snout-vent length in the s c a l i n g curves , so back ex t rapo la t ion was necessary. The 95% confidence l i m i t s of each s c a l i n g curve were p l o t t e d , and r e f l e c t i n g sample s i z e , were n e c e s s a r i l y f a i r l y wide bands. Thus, because the est imations of mass and S were based on ' w 55 small sample sizes, and values distant from the means of the sample populations, the subsequent confidence limits on the mass and S w were quite large (25-35%). Upper and lower bounds on T and T were calculated with the upper and lower limits of S w and mass respectively. Despite a l l of the assumptions, the estimate of the total propulsive force generated by H. boettgeri (see Fig. 2.10) remains positive throughout hind limb extension. In contrast, blade-element force calculations for both the angelfish and water boatman f a l l to zero or slightly negative values during the latter stages of the power stroke, when the fins/legs have completed their propulsive arc. The propulsive force and body velocity records of H. boettgeri are distinct from those of the angelfish and water boatman. It is suggested that drag-based mechanisms may not be sufficient to account for the thrust produced by swimming frogs. The caudal placement of the hind limbs indicate the potential for interactive effects. However, comparing H. boettgeri with animals that are propelled by the movement of rigid spar-like appendages, overlooks the complexities associated with multi-segment kinematics. If movement of the hind limbs is such that the maximum propulsive force per segment is staggered in time, forward thrust could be generated throughout extension. No previous attempt has been made to actually calculate the forces generated by the movement of the hind limbs of swimming frogs. A modified blade-element approach is used to calculate the forces generated by the flexion and extension of the hind limbs of H. boettgeri. Comparing this .blade-element estimate with the 56 recorded locomotor behavior, should provide information about the true nature of the propulsive forces generated by these animals. 57 CHAPTER III A BLADE-ELEMENT APPROACH TO CALCULATING THE FORCES GENERATED IN FLEXION AND EXTENSION OF THE HIND LIMBS OF HYMENOCHIRUS BOETTGERI ABSTRACT The hind limb kinematics of H. boettgeri are investigated using high speed cine films. The movement pattern is stereotypic, flexion and extension of the metatarsal-phalangeals and feet always lagging the flexion and extension of the femora and tibiofibulae. Prior to extension, the tibiofibulae are rotated, positioning the feet beyond the influence of the body and proximal hind limb segments. This is thought to be a strategy for increasing propulsive thrust, analagous to that used in competitive human swimming (the front crawl). The right hind limb was modelled as a series of linked circular cylinders and a flat plate. A blade -element approach was used to calculate the quasi-steady (drag-based) and accelerative force components parallel to the direction of motion, based on the hind limb kinematics of sequence 1 (see Chapter II). Chapter I showed that the frog's foot has similar drag coefficients to a three dimensional flat plate oriented normal to the flow. Three dimensional drag coefficients are employed for a l l hind limb segments (femur, tibiofibula, metatarsal-phalangeal, and foot). The negative thrust, generated by hind limb flexion (recovery 58 stroke) is probably responsible for the observed deceleration of the sequence 1 animal (from Chapter II, Fig. 2.7a). Positive thrust is generated only during the i n i t i a l stages of extension (power stroke), almost exclusively by the feet. The impulse of the accelerative-based thrust is substantially larger than the impulse of the drag-based thrust. Flow reversal occurs approximately midway through hind limb extension, resulting in a substantial negative thrust. However, the animal accelerates throughout extension. Hind limb interaction is thought to provide propulsive thrust for the latter half of the extension phase. It is suggested that a jet and/or ground effect may be involved. 59 INTRODUCTION Studying aquatic locomotion i s confounded by the inherent d i f f i c u l t y i n o b t a i n i n g accurate measurements of momentum exchange. General hydrodynamic models, based on known f l u i d behavior and observed system kinematics, have been developed as t o o l s to describe t h r u s t production mechanisms and corresponding p r o p u l s i v e e f f i c i e n c i e s of aquatic and semi-aquatic organisms. Using a blade-element approach, Blake (1979, 1980, and 1981a) formulated a general model f or the mechanics of drag-based p r o p u l s i o n , where p r o p u l s i v e t h r u s t r e s u l t s from the f l u i d r e s i s t a n c e to the movement of pr o p u l s i v e appendages. Blake a p p l i e d i t to p e c t o r a l f i n swimming i n the a n g e l f i s h (Pterophyllum eimekei, Blake 1979, 1980), and the water boatman [Cenicorixa b i f i d a , Blake 1986). The opportunity was taken to compare animals of two d i s t i n c t l e v e l s of o r g a n i z a t i o n ( p i s c e s and i n s e c t a ) , engaged i n hydrodynamically s i m i l a r t a s k s . Blake (1986) found that the p r o p u l s i v e e f f i c i e n c y of the a n g e l f i s h was lower than that of the water boatman. He suggested that the design of the a n g e l f i s h p e c t o r a l f i n was a compromise to f a s c i l i t a t e e f f e c t i v e p addling and hovering performance. Although a source of negative t h r u s t at the i n i t i a l and f i n a l stages of the power s t r o k e , the widened f i n base i s necessary f or the high degree of f i n ray movement a s s o c i a t e d with wave-propagation i n hovering. In c o n t r a s t , the water boatman's swimming legs are not subject to design c o n s t r a i n t s , because d i f f e r e n t mechanical tasks ( i e . pr o p u l s i o n , maneuvre, and grasping objec t s ) are performed by 6 0 d i f f e r e n t s p e c i a l i z e d limbs. Inferences regarding the e v o l u t i o n a r y b i o l o g y of organisms can often be made from what were i n i t i a l l y s t r i c t l y locomotor concepts. Few other drag-based systems have been examined i n d e t a i l . F i s h (1984) examined paddling i n the muskrat. He s i m p l i f i e d the j o i n t e d hind legs of the animal by focussing d i r e c t l y on the f o o t . He considered the proximal p i v o t of the hind femur as the centre about which the pr o p u l s i v e arc of the foot was measured. In t h i s study, a blade-element approach i s used to c a l c u l a t e the f o r c e s produced by the f l e x i o n and extension of the hind limbs of the f r o g , Hymenochizus b o e t t g e r i . Calow and Alexander (1973) c a l c u l a t e d the t h r u s t produced by the s e m i - t e r r e s t r i a l Rana temporaria, from the d e c e l e r a t i v e and a c c e l e r a t i v e phases of a v e l o c i t y record from a f l e x i o n / e x t e n s i o n c y c l e . They i n f e r r e d drag from the d e c e l e r a t i o n of the body during f l e x i o n , concluding that t h e i r r e s i s t a n c e estimate was probably an overestimate s i n c e i t was l i k e l y that hind limb r e c o i l ( f l e x i o n ) probably c o n t r i b u t e d to negative t h r u s t . Here, the hind limbs are modelled as a s e r i e s of l i n k e d c i r c u l a r c y l i n d e r s and a f l a t p l a t e . Hind limb k i n e m a t i c s , derived from high speed cine f i l m s are used together with the corresponding whole body kinematics presented Chapter I I , (from sequence 1) to c a l c u l a t e the instantaneous drag forces generated by the hind limbs during f l e x i o n and extension. Since the hind limbs do not move s t e a d i l y , the a c c e l e r a t i v e c o n t r i b u t i o n s to p r o p u l s i o n w i l l a l s o be assessed. The sum of the force (drag-based and a c c e l e r a t i v e ) components p a r a l l e l to the d i r e c t i o n of motion i s p l o t t e d as a f u n c t i o n of time throughout 61 hind limb flexion and extension. This is compared to the previous thrust estimation and iner t i a l record of the sequence 1 animal (from Chapter II, Fig. 2.10 and 2.7a respectively). Locomotor mode is often differentiated on the basis of morphological indices, such that animals with paddle (oar-like) appendages are generally catagorized as drag-based swimmers (Blake 1981, and Webb and Blake 1982). It was suggested in Chapter II that the propulsive forces generated by swimming frogs might not be wholly drag-based. Complex kinematic patterns associated with jointed hind limbs; however, could not be excluded as possible sources of variation between the whole-body velocity and propulsive force patterns of frogs (H. boettgeri and R. temporaria) and other drag-based systems (angelfish and water boatman). Calculating the instantaneous thrust produced by a l l of hind limb segments, should resolve the *jointed* kinematic issue. If this blade-element calculation alone explains the observed locomotor pattern of the sequence 1 animal, then a comparison of propulsive efficiencies (angelfish, water boatman, and frog) would be useful. If i t does not, then additional sources of propulsive thrust must be considered and assessed. Hind limb interaction, a phenomenon not considered in the basic blade-element approach, appears to play an important role in the frog system. 62 ANALYSES Frame by frame whole-body t rac ings were made from the c ine f i l m records corresponding to the four sequences introduced i n Chapter I I . The c e n t r a l long ax i s of the torso and each hind limb segment (femur, t i b i o f i b u l a , metatarsa l-phalangeal , and foot , here inaf ter designated fern, tb fb , mtph, and foot r e s p e c t i v e l y ) was est imated, and s t i c k f igures were drawn (see F i g . 3 .1) . A blade-element approach, based on the kinematics of sequence 1, was used to c a l c u l a t e the quasi-steady (drag-based) and a c c e l e r a t i v e forces generated by hind limb f l e x i o n and extens ion. The high degree of symmetry j u s t i f i e s cons ider ing the r i g h t hind limb only for the c a l c u l a t i o n s . The re levant hind limb kinematics (angular v e l o c i t i e s and angular acce le ra t ions ) were generated as fo l lows . The p o s i t i o n a l angle of each segment of the limb was measured with respect to the long axis of the torso ( F i g . 3 .2) , and p lo t t ed throughout the f l ex ion /ex tens ion c y c l e . These data were smoothed by the same technique descr ibed in Chapter II (the seven-point moving polynomial technique) to give smooth angular v e l o c i t y and angular a c c e l e r a t i o n for each segment. To c a l c u l a t e quasi-steady r e s i s t i v e forces , the limb segments were further subdivided into elements ( F i g . 3 .2 ) . Elements of the fern, tb fb , and mtph, were modelled as three-dimensional c i r c u l a r c y l i n d e r s . Foot elements were modelled as three-dimensional f l a t p l a t e s . Depending upon the r e l a t i v e v e l o c i t y of an element to the f l u i d , the r e s u l t a n t instantaneous drag force could propel or re tard the animal . Two dimensional 63 Fig. 3.1 Frame by frame whole-body tracings are made from the cine film records, as shown. The central long axes of the torso and hind limb segments are estimated, from which stick figures are drawn and positional angles are measured. 64 SNOUT 180deg n rad F i g . 3.2 The p o s i t i o n a l angles (>•) of the fern, tbfb, mtph, a n d foot segments (of the r i g h t hind limb), are measured with respect to the long axis of the torso. Cranial and caudal rotations of the segments represent increases and decreases in t h e i r p o s i t i o n a l angles, r e s p e c t i v e l y . The segments are further subdivided in to elements of s i m i l a r geometry, numbered 1-8. Element morphometries are given in Table 3.1. Refer to ANALYSES for further explanation. 65 vector geometry allowed the calculations of relative velocity and associated resistive forces on each element. Fig. 3.3 illustrates an example of how the component of the drag-based force on an element, parallel to the direction of motion, was calculated. Consider element i , located at the distal end of segment j . The distance from the midpoint of element i to the pivot point of segment j , is r. The pivot point of segment j has velocity v n e t -The midpoint of element i has a tangential linear velocity of ru> (where o> is the angular velocity of segment j ) . v n e*. opposes rco, such that the velocity of element i , relative to the fluid is v, which Is given by, v=V(v 2+ v 2) (3.1) n s where v =r<o+V . siny and v =V . cosy (3.2) n net s net and r is the positional angle of segment j . The drag force dF y, opposing the relative velocity v, is given by, dF y=l/2pS wC Dv 2 (3.3) where p, S w, and Cp are fl u i d density, projected area, and drag coefficient respectively. The component of this resistive force parallel to the direction of motion, dT, is 66 Fig. 3.3 The drag-based thrust (dT) of element i , located on segment j , is calculated. Refer to ANALYSES, equations 3.1-3.8, for further explanation. 67 dT=dF cos(x) (3.4) where x=180°-y°-e° (rcrad - ^ r a d - e r a d ) , and & i s the hydrodynamic angle of a t t a c k of element i . © i s given by, ©=arcsin(v /v) n (3.5) The areas of the elements are given i n Table 3.1 with other blade-element morphometries. C y l i n d r i c a l ( f o r fern, t b f b , and mtph elements) areas were given by the product of t h e i r l e n g t h and diameter. The p r o j e c t e d areas of the f o o t elements were measured d i r e c t l y from the c i n e f i l m . F i l m frames immediately preceding sequence 1 showed the animal r o t a t i n g i n the plane of -the camera, where the expanded f o o t c o u l d be a c c u r a t e l y measured. The drag c o e f f i c i e n t s of the f o o t elements are f u n c t i o n s of t h e i r hydrodynamic angles of a t t a c k , where k=2.5 (from Blake 1979). The drag c o e f f i c i e n t s of the c y l i n d r i c a l elements are a l s o f u n c t i o n s of t h e i r hydrodynamic angles of a t t a c k , C n = l . l f o r ©±40°(0.70rad) from 90° (1.57rad) (3.6) (3.7) C n = l . l s i n ©+0.02 (3.8) 68 TABLE 3.1 HIND LIMB ELEMENT MORPHOMETRICS Element Element Element- Element Added V i r t u a l Number Desc r ip t ion Area m Mass kg. Mass kg. Mass kg. 1 fern (proximal) 4 . 5 2 x l 0 " 6 2 fern ( d i s t a l ) 4 . 5 2 x l 0 - 6 3 tbfb (proximal) 2.30x10" 6 4 tbfb ( d i s t a l ) 2 .30xl0~ 6 5 mtph 2 . 3 8 x l 0 " 6 6 foot (proximal) 4 . 1 7 x l 0 - 6 l . O x l O - 6 6. 93x10" •6 7. 93x10" •6 7 foot (middle) 1 .25x l0 " 5 3 . 0 x l 0 " 6 6. 14x10" •5 6. 44x10" •5 8 foot ( d i s t a l ) 5 . 2 1 x l 0 " 6 1 . 2 5 x l 0 - 6 3. 27x10" 5 3. 40x10" 5 69 (Hoerner 1965) for subcritical Reynolds Numbers (ie. R < 5.0x10 ). e For the purpose of this analysis, the direction of the forward velocity of the animal (Vg, 'vent' velocity, see Fig. 2.6a, Chapter II) is always positive. The directions of angular velocities of the hind limb segments are therefore, positive during flexion, and negative during extension. A propulsive force is generated by an element only when i t is effectively moving backward (negative relative velocity, v). Because they are linked, proximal segments influence the velocities of distal ones. V n e t describes the velocity that any segment *sees f i t s e l f travelling at, as a result of being attached to other moving bodies. v n e*. * s segment specific and is equal to Vfi for the fern segment. For the foot segment, V ^ Is equal to the sum of Vg and a l l of the velocity contributions parallel to the direction of motion, of the more proximal segments. v n e*- D e negative or positive, for the proximal and dis t a l segments respectively, during extension. v n e*. * s always positive for every segment during flexion. Specific vectors were adjusted according to the particular element being considered. The total drag-based force generated by the right hind limb, parallel to the direction of motion, is therefore equal to the sum of the dT contributions of a l l of the elements throughout flexion and extension. This value is doubled to give the total for both hind limbs as follows, e=8 dT tot' E dF ycos(x) e=l (3.9) 70 The forces associated with the unsteady movement of the legs must be considered. When a body is accelerated through a f l u i d , a volume of fluid is entrained with i t . The force required to accelerate this body therefore, must be of sufficient magnitude to accelerate the mass of the body and the mass of the volume of flu i d entrained (the added mass). This added mass depends on the size, orientation, and type of motion that the moving body experiences (translational, rotational etc.). The reaction force (acceleration reaction, Daniel 1984) associated with accelerating the virtual mass (mass plus added mass) of an element, is in a direction opposite to that of the acceleration (+ or - ) . There are undoubtedly unsteady forces associated with a l l of the hind limb segments; however, only those of the foot elements are considered in this analysis. The foot experiences the highest angular accelerations. Being the most dist a l segment, i t is also experiences the greatest translational acceleration as a result of the contributions of the extending proximal segments. The example in Fig. 3.4, shows how two dimensional vector geometry is used to solve for these accelerative forces. Consider element i , located at the di s t a l end of segment j . The distance from the midpoint of element i to the pivot point of segment j is r. During extension, segment j has an angular acceleration a, the direction of which opposes the direction of progression of the body (V f i). Element I has a linear tangential acceleration ( Af- a n) of ra.. Element i also has a centrepetal 2 acceleration ( A ) of ( r t o ) / r (where co is the angular velocity of cen •* segment j ) . The resultant acceleration of element i Is given by, 71 Fig. 3.4 The accelerative force ( F a d d ) of element i , located on segment j , is calculated. Refer to ANALYSES, equations 3.10-3.14, for further explanation. 72 A =V(A. 2+A 2) (3.10) res tan cen Adding A . to A gives A the relative acceleration of net res rei element i to the f l u i d . A . is analagous to V . (in the net 3 net drag-based analysis), as i t represents the sum of the acceleration of the body of the animal (vent acceleration of Sequence 1, Chapter II, Fig. 7a), and the parallel components of the accelerations of the more proximal segments (mtph, tbfb, and fern). A , is calculated by, rei A 2=A 2+A 2-2A .A cosQ (3.11) rei net res net res where, Q=y+T {y =positional angle of segment j) , and T=asin(A. /A ). The acceleration reaction of element i is tan res proportional to the normal component of A r e^, A =A .cosP (3.12) n rei where P=90O-S°-T° (n/2rad-Srad-Trad) and S=arcsin((A .sinQ)/A , ) . The componenet of the acceleration net r ei reaction parallel to the direction of motion (F,^) * s proportional to A^^, A a d d=A ncos(0) (3.13) where 0=90°-^° (Tr/2rad-yrad). F g ( J ( 3 is given by, 73 add~ mvirtual^add (3.14) where m virtual is the mass of the element plus i t s associated added mass. The added masses of the foot elements were generated by solving for the mass of the volume of fl u i d described by rotating rectangular-area equivalents of each element about the long axis of the foot segment (see Table 3.1 for virtual masses of the foot elements). In this example, F ^ is propulsive (ie. in the same direction as the velocity of the body, Vg). The total unsteady force generated by the foot during extension only, is the sum of a l l F of each foot element, add e=8 addtot =2 £ m virtual add (3.15) e=6 74 RESULTS Symmetric composite stick diagrams, illustrating the positions of the hind limb segments relative to the long axis of the torso, are shown in Fig. 3.5 a and b (during flexion 1-5, and extension 6-10, respectively). This stereotypic pattern is characterized by the following five stages, beginning with Fig. 3.5a-l. 1. Flexion of ferns and tbfbs, mtphs and feet maintained parallel to the long axis of the torso. 2. Flexion of mtphs and feet, feet folded and webbing collapsed. 3. Extension of tbfbs to place feet well beyond the knees. 4. Extension of ferns and tbfbs , mtphs and feet maintained normal to the long axis of the torso. 5. Extension and rotation of mtphs and feet. Flexion and extension of the mtphs and feet always 'lag' the flexion and extension of the ferns and tbfbs. The positional angles recorded for the right hind limb segments of sequence 1 are shown in Fig. 3.6. Flexion and extension are defined in terms of the movements of the foot segment. The positional angles of the fern, mtph, and foot, increase approximately in phase to a maximum at the end of flexion. Tbfb, however, shows an decrease in positional angle during the latter stages of flexion, corresponding to the latteral movement of the feet (stage 3 above). There is essentially an decrease in positional angle throughout extension ln a l l segments. Mtph and the foot continue to decrease in positional angle, beyond 75 Fig. 3.5 A symmetric composite stick diagram of the torso and hind limb segments of H. boettgeri is shown. The numbers 1-5 and 6-10 represent sequential orientations throughout flexion (recovery phase) and extension (power phase), respectively. The snout-vent length \, is shown. V"B indicates the body (vent) velocity of the animal. 76 z < Q < CC LU _ l O z < < z o to o CL 1.2 iu4> 00 1.6-i 1.2 0.8 -! OX I oo 1.6 1.2-1 0.8 0.L H 0.0^ 1.6-1.2 0.8 OA 0.0®-.•sets 8* 8, ocP 0 '88 t§8 o • o O O l oo -FLEX-0.10 020 TIME SEC I - 4 - E X T - J 030 Fig. 3.6 The experimental ( O ) and smooth ( • ) positional angles of the right hind limb segments (fern a, tbfb b, mtph c, and foot d) are shown throughout the hind limb flexion and extension of the sequence 1 animal. Refer to ANALYSES for further explanation. 77 the stages of extension 'proper', because the fern and tbfb begin to flex again. The foot shows the most pronounced change in positional angle. Smooth values parallel experimental ones. The angular velocity of each segment is shown in Fig. 3.7. Negative values indicate extension. The foot experiences the greatest angular velocities during both flexion and extension. The angular acceleration of each segment is shown in Fig. 3.8. Segments accelerating and decelerating during extension w i l l have negative and positive values, respectively. Peak accelerations of the fern, mtph, and foot segments are approximately ln phase during extension. The foot reaches a -2 maximum angular acceleration of about 1200 radsec during the i n i t i a l stages of the the extension phase. The relative velocities of five of the eight hind limb elements (elements 2, 4, 5, 6, and 8 see Fig. 3.2 and Table 3.1), calculated by the method outlined In Fig. 3.3, are shown in Fig. 3.9. A negative value reflects a *backward' movement. Elements 2 and 5 experience a brief negative velocity late in flexion and early in extension respectively. The foot elements 6 and 8 (and element 7, not shown) move backwards approximately throughout the f i r s t half of the extension phase. During the latter half of extension, the rearward velocity of these elements is overcome by the forward velocity of the body. Elements 1, 2, and 3 (1 and 3 not shown) never have a rearward velocity great enough to supercede the forward velocity of the body. The total force generated by both hind limbs is shown in Fig. 3.10. The drag-based component is proportional to the relative 78 Fig. 3.7 The smooth angular velocities of the right hind limb segments (fern , tbfb - - - • , mtph — , and foot ) are shown, throughout the hind limb flexion and extension of the sequence 1 animal. 79 Fig. 3.8 The smooth angular accelerations of the right hind limb segments (fern , tbfb , mtph , and foot HBB^B) are shown,throughout the hind limb flexion and extension of the sequence 1 animal. 80 Fig. 3.9 The relative velocity (v, see Fig. 3.3) of the right hind limb elements (2 ,4 — '—,5 ,6-—— , and 8 see Fig. 3.2) are shown throughout the hind limb flexion and extension of the sequence 1 animal. A negative relative velocity represents a 'backward' movement of the element. 81 F i g . 3.10 The t o t a l drag-based force ( d T t o t from equation 3.9) i s shown (— ), throughout the hind limb f l e x i o n and extension of the sequence 1 animal. The t o t a l a c c e l e r a t i v e force ( F a d d f r o m equation 3.17) is shown ( ), throughout hind limb extension. The sum of the drag-based and accelerative forces ( ) i s shown. Refer to ANALYSES and DISCUSSION for further explanation. 82 velocities of a l l of the right hind limb elements during flexion and extension (except the foot elements during flexion). The accelerative component is derived from the relative accelerations of the right foot elements, only during extension. Flexion of the hind limbs generates negative thrust. The net positive thrust produced during the f i r s t half of the extension phase is almost completely accelerative. The drag-based forces generated during this phase are negligible by comparison. At about the midpoint of the extension phase, the forward velocity of the body has Increased, causing flow reversal and the production of negative drag-based thrust over the latter half of hind limb extension. The deceleration of the hind limb segments during the latter stages of extension, contribute to the generation of a substantial negative accelerative impulse, about 2.5 times that of the corresponding negative drag-based impulse 83 DISCUSSION The pattern of movement illustrated in Fig. 3.5 is stereotypic. During flexion, the mtphs and feet are always drawn up behind the ferns and tbfbs and may be shielded, reducing resistance during hind limb flexion. Prior to extension 'proper', the tbfbs are rotated to position the feet l a t t e r a l l y beyond the knees. This may be a strategy for maximizing propulsive thrust, analagous to that employed by competitive swimmers. The propulsive phase of the front crawl involves drawing the arm in an 'S' pattern, below the body. Forward propulsion through fluids is maintained by giving the fluid rearward momentum. The 'S' pattern is an attempt to interact with the greatest volume of . undisturbed f l u i d . The more fluid one can interact with, the greater the momentum exchange and the greater the propulsion. Despite streamlining, the fluid flowing over the frog's body wi l l be somewhat disturbed. Latteral placement of the foot by the rotation of the tbfbs, may enable the animal to interact with the relatively undisturbed bulk flow. This would translate into a greater propulsive effort for the animal. The components of the drag-based and accelerative-based forces, parallel to the long axis of the frog's body, have been calculated on the basis of the hind limb kinematics of sequence 1, a complete flexion/extension cycle. The assumptions involved in this calculation should be considered. Only the right hind limb kinematics were used directly, and a l l final force values were doubled to account for both legs. Since sequences were 83 DISCUSSION The pattern of movement illustrated in Fig. 3 . 5 is stereotypic. During flexion, the mtphs and feet are always drawn up behind the ferns and tbfbs and may be shielded, reducing resistance during hind limb flexion. Prior to extension 'proper 1, the tbfbs are rotated to position the feet l a t t e r a l l y beyond the knees. This may be a strategy for maximizing propulsive thrust, analagous to that employed by competitive swimmers. The propulsive phase of the front crawl involves drawing the arm In an 'S' pattern, below the body. Forward propulsion through fluids is maintained by giving the fluid rearward momentum. The *S' pattern is an attempt to interact with the greatest volume of . undisturbed f l u i d . The more fluid one can interact with, the greater the momentum exchange and the greater the propulsion. Despite streamlining, the fluid flowing over the frog's body wi l l be somewhat disturbed. Latteral placement of the foot by the rotation of the tbfbs, may enable the animal to interact with the relatively undisturbed bulk flow. This would translate into a greater propulsive effort for the animal. The components of the drag-based and accelerative-based forces, parallel to the long axis of the frog's body, have been calculated on the basis of the hind limb kinematics of sequence 1, a complete flexion/extension cycle. The assumptions involved in this calculation should be considered. Only the right hind limb kinematics were used directly, and a l l final force values were doubled to account for both legs. Since sequences were 84 specifically chosen for hind limb symmetry, this is a valid simplification. The movement of the hind limbs throughout flexion and extension was coplanar with the long axis of the animal's body, such that two dimensional vector geometry is sufficient to calculate resultant forces. The hind limb is modelled a series of linked three dimensional circular cylinders and a flat plate, in free flow (fern, tbfb, mtph, and foot respectively). These are further divided into elements, smaller units of similar geometry. The elements were assigned drag coefficients corresponding to their technical equivalents. Previous blade-element modelling studies (Blake 1979, 1986 and Fish 1984) have assigned force coefficients to the elements of the swimming appendages (the pectoral fin of the angelfish, the swimming leg of the waterboatman, and the foot of the muskrat, respectively), based on the f l a t plate analogy. Blake points out that i t is likely that only the outermost element experiences anything like 'free flow', but concluded that since this element was responsible for about 90% of the propulsive thrust, the imput from inbound elements was relatively unimportant. In Chapter I free f a l l drag studies with a frog's foot and acetate models thereof, gave drag coefficients similar to those derived from experiments with dynamically similar three-dimensional flat plates, normal to the free flew (0^=1.1, Hoerner 1965). Perhaps more importantly, the 'subtraction method' results from Chapter I, show, at least at higher Reynolds numbers (>1000), that the foot behaves like a flat plate in free flow, even when i t is attached to the limb (C^ also about 1.0). These results justify using three dimensional flat plate equivalents to 85 describe the hydrodynamic properties of the frog's foot, and support previous modelling studies involving propulsion by rigid spar-like appendages. However, the frog's jointed legs introduced a level of complexity not involved in previous studies. Empirical data for the fern, tbfb, and mtph, analagous to that obtained for the foot, is lacking. It was necessary to employ values for drag coefficients from the literature. The kinematic pattern observed in swimming frogs, suggests interaction between the segments is lik e l y , particularily during flexion. Shielding may be important in reducing the resistance to flexion. Throughout this calculation, a l l segments were treated as though they were functionally isolated. The only interaction considered was the extent to which proximal segment velocities and accelerations affected more distal segments. The calculated drag-based forces, particularly during flexion, are probably overestimates. The accelerative-based force calculations during extension, are more speculative. Added mass coefficients, sensitive to shape and size, also depend on the type of motion considered (translational, rotational etc.). The foot experienced both translational (due to the extension of proximal segments) and rotational (about the mtph) motion. Here, the added masses of the foot elements are based on translational motion. Additionally, this acceleration reaction and its coeffiecient are derived from ideal flow theory, which admits no vorticity. When vortex formation and shedding occurs, analytical solutions for added mass coefficients are more d i f f i c u l t to derive. Birkhoff (1960) suggests that ideal flow theory is applicable within the f i r s t 86 three diameters of travel. The added masses of the foot elements were derived from element volumes of rotation (Blake 1979) and give reasonable estimates. Velocities and accelerations of the hind limb segments were smoothed by the same smoothing technique as that introduced in,Chapter II. The merits of this technique have been discussed (see RESULTS, Chapter II), such that the relative magnitudes and directions of the calculated forces, and therefore the overall force record probably gives a r e a l i s t i c view of the dynamics of this jointed system. Fig. 3.10 shows that positive (propulsive) forces are generated only during the f i r s t half of the extension phase. The feet are largely responsible for this by virtue of their large area and relatively high rearward acceleration (see Fig. 3.8) The accelerative impulse far exceeds the drag-based impulse during the i n i t i a l period of extension. The small element areas, coupled with low relative velocities translates to small drag-based forces. The retarding drag-based forces generated by the proximal elements (see Fig. 3.2) essentially cancel the propulsive drag-based forces of the foot elements, such that the sum of the drag-based contributions of a l l of the elements during the i n i t i a l stages of extension is zero. Alternatively, the acceleration of the foot segment is quite high (see Fig. 3.8), and coupled with large element added masses (see Table 3.1), the accelerative impulse is substantial. It has been suggested that accelerative-based thrust can be an important component of total thrust, i f the stroke axis is asymmetric. Nachtigall (1960) found 87 that both the stroke angle and axis of rotation of dytiscid beetles is approximately 2/3U radians, and suggested that they move their swimming legs in a way that would maximize thrust i f the acceleration reaction was a dominant source of thrust. Daniel (1984) showed that paired synchronous limbs can generate a maximum accelerative force i f the stroke angle and axis of oscilation is 2/3U radians. Accelerative thrust production by the related [Cenicozixa bifida, Blake 1986) accounted for about one third of the total thrust produced. Stroke angles and axes of rotation were similar to the dytiscid beetles investigated by Nachtigall. Though their absolute magnitudes are often d i f f i c u l t to determine, accelerative-based forces can be important sources of propulsive thrust for animals. As the animal begins to accelerate and i t s velocity increases, the relative 'backward' velocities and accelerations of the hind limb elements diminish. Additionally, any drag-based or accelerative-based forces that are generated during the final stages of extension are lat t e r a l l y directed and essentially cancel. The result of flow reversal and force cancellation is a negative (retarding) thrust, throughout the latter half of hind limb extension. This force calculation, is markedly different from the propulsive thrust (T and T), predicted from the force balance (Fig. 2.10, Chapter II). The range of T and the iner t i a l requirement, given the pattern of acceleration observed ln sequence 1, are compared to the blade-element force calculation in Fig. 3.11. Drag-based and accelerative forces can only support the acceleration of the animal's body during the i n i t i a l stages of 88 to ' o 2.51 2.0-1.5 H x 1.0 LU 0.5 -o rr o 0.0 -0-5 H -1.0 002 0.06 0.10 TIME SEC. FLEXION 0.14 0.18 EXTENSION-Fig. 3.11 The total drag-based and accelerative force ( from Fig. 3.10, Chapter III), is compared to F n g ^ ( — — — — from Fig. 2.10, Chapter II), throughout the hind limb flexion and extension of the sequence 1 animal. The shading indicates the range of T (from Fig. 2.10, Chapter II). The stick figures represent the approximate position of the hind limbs at the beginning, midpoint, and end of the extension phase. Refer to DISCUSSION for further explanation. 89 extension. The fact that the animal continues to accelerate, despite corresponding calculations of negative thrust, suggests that an alternative source of propulsive thrust is operating in this system. The transition from positive to negative thrust is concommitant with the hind limb orientation shown in Fig. 3.11. Following the extension of the ferns and tbfbs, the mtphs and feet begin to extend and rotate towards each other. The force calculation was based on the right hind limb only, treated as a functionally isolated unit operating in free space, so potential interactive effects between limbs were not considered. It is suggested that an additional force, generated by hind limb interaction, is responsible for the animal's acceleration during the latter half of the extension phase. Two possible, mechanisms come to mind. The f i r s t , a *jet' effect, results from the redirection of the fluid between converging surfaces. Distinct from jet propulsion, a locomotor mode associated with the contraction of discrete f l u i d - f i l l e d 'sacs', the jet effect manifests Itself whenever surfaces meet. It Is l i k e l y to be important in aquatic systems where propulsion is achieved by the movement of multiple, closely-packed appendages (eg. errant polychaete worms). The second mechanism, known as a reflective effect, is less well understood. It is known that i f two surfaces are brought close together, the total f l u i d force is greater than the sum of the forces generated by each surface in free flow. It is analogous to the ground effect, where the l i f t of an aerofoil is enhanced by proximity to a surface (Reid, 1932). A number of observations support the notion of jet and/or 90 ground e f f e c t s . The d i s p a i r i t y between the c a l c u l a t e d and r e q u i r e d f o r c e s (see F i g . 3.12), occurs when the hind limbs can p h y s i c a l l y i n t e r a c t . Hind limb i n t e r a c t i o n c o u l d not e x p l a i n such a d i s p a i r i t y i n the e a r l y stages of e x t e n s i o n . Of the four sequences examined i n Chapter I I , three of them i n d i c a t e d that animals were a c c e l e r a t e d throughout hind limb e x t e n s i o n (Chapter I I , F i g . 2 . 7 ) . F i g . 2.7b (Chapter II) shows t h a t the animal experiences an a d d i t i o n a l a c c e l e r a t i o n i n a time frame c o n s i s t a n t with the i n t e r a c t i o n and r o t a t i o n of the mtph and f e e t . Support f o r i n t e r a c t i v e e f f e c t s a l s o come from o b s e r v a t i o n s of H. b o e t t g e r i d u r i n g slow swimming. The f o r e limbs, t o r s o , and femoral and t i b i o f i b u l a r hind limb segments, remain q u i t e r i g i d . The body i s p r o p e l l e d o n l y by the r o t a t i o n a l movements of the mtphs and f e e t . In slow swimming, the p o s i t i o n a l angles of the f e e t are such t h a t drag-based f o r c e s would be p o o r l y d i r e c t e d . However, as the expanded, o f t e n bowed f e e t r o t a t e , they c o u l d squeeze water backwards. Although the 'sac' i s l e s s w e l l d e f i n e d , the combined m o b i l i t y of the toes and f l e x i b l e webbing could a l l o w more p r e c i s e volume and d i r e c t i o n c o n t r o l . T h i s would minimize i n e f f i c i e n c i e s a s s o c i a t e d with 'leakage'. I t would seem t h a t a combination of r e a c t i v e , r e s i s t i v e , and I n t e r a c t i v e e f f e c t s p r o p e l the f r o g . A c c e l e r a t i v e f o r c e s appears to be the dominant source of p o s i t i v e t h r u s t over the f i r s t h a l f of e x t e n s i o n . I n t e r a c t i v e f o r c e s (most l i k e l y j e t f o r c e s ) , appear to be Important In m a i n t a i n i n g a c c e l e r a t i o n of the animal throughout the l a t t e r h a l f of e x t e n s i o n . The f o r c e - g e n e r a t i n g mechanisms addressed here are not mutually e x c l u s i v e . Within a 91 s i n g l e l o c o m o t o r s y s t e m , s e v e r a l mechanisms may f u n c t i o n i n c o n c e r t t o produce o b s e r v e d movement p a t t e r n s . C e r t a i n l y t h e r e a r e dominant mechanisms, but c a t a g o r i z i n g t h e l o c o m o t o r b e h a v i o r of o r g a n i s m s on t h i s b a s i s c o u l d r e s u l t i n t h e n e g l e c t of o t h e r i m p o r t a n t , a l b e i t l e s s o b v i o u s mechanisms. 92 CONCLUDING COMMENTS The hydrodynamic drag of the body and hind limbs the a q u a t i c f r o g Hymenochirus b o e t t g e r i are s i m i l a r (as r e f l e c t e d by the drag c o e f f i c i e n t , Cp) t o the s e m i - t e r r e s t r i a l Rana p i p i e n s (Chapter I ) . C l e a r l y a comprehensive I n v e s t i g a t i o n of a spectrum of anuran body morphology would be necessary to draw any sound c o n c l u s i o n s about a d a p t a t i o n s t o the s p e c i f i c locomotor modes employed by these animals. For example, i f a lower s u r f a c e area to volume r a t i o (a more s p h e r i c a l , l e s s s t r e a m l i n e d shape) was an important a d a p t a t i o n f o r m i n i m i z i n g water l o s s In the more t e r r e s t r i a l forms, drag experiments analagous t o those i n Chapter I, would i n d i c a t e a l e v e l of compromise by a r e l a t i v e l y higher drag c o e f f i c i e n t . Anurans d i s p l a y an extreme range of l i f e h i s t o r y s t r a t e g i e s . Some engage In the r a t h e r u n c h a r a c t e r i s t i c h a b i t of l a y i n g eggs on land (Rana gray! and Rana fasciata f o r example, F r a s e r 1973). These animals are such poor swimmers t h a t they w i l l a c t u a l l y drown i n deep water. I t Is i n t e r e s t i n g t o sp e c u l a t e about the types of t e r r e s t r i a l a d a p t a t i o n s t h a t would so compromise swimming performance. General hydromechanical models r e l y on s i m p l i f y i n g assumptions as a means of e x p l o r i n g broad t r e n d s i n animal p r o p u l s i o n . P r e v i o u s m o d e l l i n g s t u d i e s have c o n s i d e r e d the p r o p u l s i v e appendages of p a d d l i n g animals t o be hydrodynamically analagous t o three dimensional f l a t p l a t e s , o r i e n t e d normal t o the d i r e c t i o n of flow. D e s p i t e s u b s t a n t i a l d i f f e r e n c e s i n s u r f a c e t e x t u r e , the f r e e f a l l , s u b t r a c t i o n , and flow v i s u a l i z a t i o n 93 results with the foot of H. boettgeri (Chapter I) support this. A substantial portion of the discussion of Chapter II deals with the problem of determining the drag experienced by a flexible body. The static drag coefficient of the whole-body and splayed hind limbs of H. boettgeri (measured in Chapter I), gave a poor estimate of the instantaneous resistance of the streamlined animal (Fig. 2.5). A range of drag coefficients corresponding to the series of hind limb orientations, is cited as a possible method for determining the resistance experienced by the swimming animal. However, dynamic resistance continues to be a d i f f i c u l t conceptual and experimental issue in blofluiddynamics. The results of Chapter II and III suggest that a combination of reactive, resistive and Interactive forces propel ffymenochirus boettgeri and probably other frogs. The interactive effects described in Chapter III are of particular interest, because they suggest the potential for active optimization of fl u i d propulsion systems, by the selection of specific movement patterns or •gaits'. This idea is supported by the fact that H. boettgeri has never been observed to swim by asynchronous hind limb extensions, at any speed. Hind limb asynchrony is only associated with directional changes. While slow swimming in the angelfish is accomplished exclusively by asynchronous pectoral f i n rowing, water boatmen are always propelled by the synchronous rowing actions of their hind legs (Blake 1979 and 1986 respectively). Appendage morphology of paddlers tends toward the optimum triangular shape predicted by Blake (1981) on theoretical grounds. Although a broad convergence of appendage morphology is evident, 94 paddling animals display considerable v a r i a b i l i t y in appendage movement patterns. Investigating the hydrodynamic consequences of variable stroke patterns would be a worthwhile endeavor. 95 LITERATURE CITED ALEXANDER, R.McN. 1971. Size and Shape. London, Edward Arnold. BAIRD, T. 1983. 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