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The biomechanics of jellyfish swimming Megill, William MacDonald 2002

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T H E BIOMECHANICS OF JELLYFISH SWIMMING by William MacDonald Megill B.Sc. (Physics) McGill University, 1991 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In THE FACULTY OF GRADUATE STUDIES Department of Zoology We accept this thesis as conforming to the required standard The University of British Columbia June, 2002 © W I L L I A M M . M E G I L L , 2002 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 Vancouver, Canada Date DE-6 (2/88) Abstract Many jellyfish produce forward thrust by ejecting a volume of water contained within their subumbrellar cavities. This behaviour is powered by a single set of muscles lining the inside of the cavity, and controlled by a simple nerve ring at the base of the mesogleal bell. Refilling is powered by strain energy stored in the deformation of the bell. Despite the apparent simplicity of the system, four gait patterns can be identified in the swimming behaviour of the hydrozoan jellyfish, Polyorchis penicillatus Eschscholz. In increasing order of activity, the behaviours are: hopping, sink-fishing, transient swimming, and resonant swimming. The first two are used in fishing, while die others are active swimming gaits. Thrust is produced in the same way in all gaits, but in the highest frequency gait, the animal can benefit from the phenomenon of resonance to produce additional thrust. The mesoglea is a fibre-reinforced mucopolysaccharide gel. The fibres have recently been identified as microfibrils made of a protein homologous to mammalian fibrillin. Most of the energy required to power the refill stroke is thought to be stored in the stretch of these microfibrils. The elastic modulus of microfibrils has been measured in other systems and found to be in the range of 0.2 to 1.1 MPa. Jellyfish microfibrils are found here to have a similar stiffness of approximately 0.40 MPa, more than enough to account for the energy necessary to refill the bell cavity. Additional energy storage is found to be available in the deformation of the mesogleal matrix itself - the stiffness of the joint mesogleal matrix is found to be approximately 130Pa, while that of the bell mesoglea is found to be approximately 352Pa. The elastic behaviour of the whole system is non-linear, both in deflation and inflation, but can be modelled as a thick-walled cylinder made of a fibre-reinforced composite material. A numerical model of this geometry is developed and used to correctly predict the observed quasi-static behaviour of a mounted jellyfish preparation. The dynamic elastic behaviour of the jellyfish bell is modelled as a non-linear oscillator. The model successfully predicts the kinematics of the mounted preparation, matching the resonant frequency using measurements and reasonable estimates of the input parameters. It makes the additional prediction that the resonant frequency should be a function of the forcing amplitude. The model and mounted preparation are found to resonate at frequencies much higher than the free-swimming animals. Once the boundary conditions are adjusted to reflect the geometry of the free-swimming animal, the effective stiffness of the bell is found to be lower, and the model then predicts a lower resonant frequency, closer to the free-swimming frequency. Ill The swimming frequency of the jellyfish depends on the stiffness of its bell and on the mass of water entrained by its motion. During refilling, a toroidal vortex is set up as water displaced by the expanding bell wall is sucked into the bell cavity through the velum. This mass, coupled with a measurement of the damping due to hydrodynamic drag and the adjusted stiffness, is used with the nonlinear oscillator model to predict the swimming frequency of the resonant gait for Polyorchis of all sizes. iv Table of Contents Abstract 1 1 Table of Contents i v List of Tables v i List of Figures vii List of Symbols l x Acknowledgements x Chapter 1. General Introduction 1 Jellyfish Locomotion 1 The Study Animal 2 Ecology 2 Physiology 4 Structure of Thesis 1 1 Chapter 2. Kinematics, Gaits, and Resonance 13 Introduction 13 Materials & Methods 1 5 Results 1 8 Morphology 1 8 Gaits 1 8 Kinematics 24 Discussion 27 Muscular control 27 Harmonic oscillator 29 Chapter 3. The Modulus of Elasticity of Jellyfish Microfibrils 34 Introduction 34 Materials & Methods 35 Study animals 35 Microscopy 35 Mechanical testing 37 Results 4 0 Fibre morphology 40 Fibre densities 43 Fibre diameters 43 Mechanical properties of mesoglea 47 Discussion 51 V Chapter 4. From Mesoglea to Jellyfish.... 5 4 Introduction : • 4^ Model 55 Transversely isotropic composite material 55 Jellyfish geometry • 60 Materials & Methods 61 Study animals 61 Apparatus 62 Calibration 6 4 Results : • : 6 4 Pressure-volume data 64 Fitting the model 67 Discussion 67 Chapter 5. Dynamic Elastic Behaviour (Resonance) 7 1 Introduction 7i The Model 7 2 Resonance in a simple harmonic oscillator 7 4 Non-linear oscillator model 7 ^ Materials & Methods 7 9 Study animals 7 9 Apparatus 8 * Calibration 8 1 Testing protocol 8 ^ Results 8 5 Spring restoring force 8 ^ Kinematics, mass & damping 8 8 Resonant frequencies 8 8 Discussion 9 4 The free-swimming animal 94 Conclusion m 5 Chapter 6. General Discussion 107 References I* 2 vi List of Tables Table 2.1. Relative importance of jellyfish gaits 22 Table 2.2. Fit parameters for Figure 2.9 32 Table 3.1. Fibre densities 45 Table 3.2. Fibre diameters 46 Table 3.3. Elastic moduli of jellyfish mesoglea in radial tension 48 Table 4.1. Third order polynomial fits to the pressure-volume data in Figure 4.4 66 Table 4.2. Stiffnesses and critical volume parameters required to fit the model to experimental data 69 Table 5.1. Fit coefficients 87 Table 5.2. Masses, damping, and resonant frequencies 92 vii List of Figures Figure 1.1. The penicillate jellyfish, Polyorchis penicillatus. 3 Figure 1.2. Polyorchis morphology 5 Figure 1.3. Strain distribution during deflation 7 Figure 1.4. Stress-strain behaviour of mounted jellyfish preparation 8 Figure 1.5. Contraction timing of Polyorchis swimming muscle contraction 10 Figure 1.6. Flowchart of thesis structure 12 Figure 2.1. Morphological measurements. 16 Figure 2.2. Fineness ratio as a function of size 19 Figure 2.3. Damped oscillations in transient swimming 20 Figure 2.4. Kinematics of the resonant gait 23 Figure 2.5. Jellyfish swimming frequencies in active gaits 25 Figure 2.6. Pulse shape variability 26 Figure 2.7. Reynolds numbers 28 Figure 2.8. Muscle activation and contraction timing during jet phase 30 Figure 2.9. Damped harmonic oscillator - fit to data of linear vs. non-linear models 32 Figure 3.1. Polyorchis morphology 36 Figure 3.2. Mechanical testing apparatus 38 Figure 3.3. Collage of micrographs showing a complete cross-section of a jellyfish 41 Figure 3.4. Micrographs used to measure the density of jellyfish microfibrils 45 Figure 3.5. Typical micrographs used to measure fibre diameters 46 Figure 3.6. Stress-strain curve of an isolated mesoglea sample 49 Figure 3.7. Typical stress-strain behaviour of an intact animal preparation 50 Figure 4.1. Jellyfish geometry - thick walled cylinder with radial fibres & longitudinal joints 56 Figure 4.2. Transversely isotropic thick-walled cylinder model 57 Figure 4.3. Experimental apparatus 63 Figure 4.4. Quasi-static pressure-volume behaviour of the mounted jellyfish preparation 65 Figure 4.5. Experimental data and fit of transversely isotropic thick-walled cylinder model 68 Figure 5.1. Resonance behaviour of a linear harmonic oscillator 73 Figure 5.2. Non-linear resonance (Duffing's equation) 76 Figure 5.3. Solution to the non-linear oscillator 80 Figure 5.4. Volume transfer function 82 Figure 5.5. Pressure transfer function 84 Figure 5.6. Force-extension data 86 Figure 5.7. Kinematics of two typical experiments 89 Figure 5.8. Experimental verification of non-linear resonance 93 Figure 5.9. Free-swimming spring restoring force 97 Figure 5.10. Muscle force Figure 5.11. Circular vortex model of entrained mass Figure 5.12. Mass of entrained circular vortex model as a function of jellyfish bell height. Figure 5.13. Recoil frequency decreases as the square root of bell height Figure 5.14. Benefit of resonance Figure 6.1. Pulse kinematics, squid vs. jellyfish List of Symbols Symbol Use Symbol Use Ai s longitudinal section area spring function fit coefficients A s sample cross sectional area lo fibre length, sample length A u subumbrellar area m mass a mass coefficient Q oscillator quality (mounted preparation) b damping coefficient v Poisson's ratio, isotropic material B oscillation amplitude V T L , V L T Poisson ratio, fibre plane P mass coefficient V T T Poisson ratio, transverse plane (free-swimming) 5 phase shift n fibre density e strain P pressure Ef fibre strain ro resting margin radius matrix strain r instantaneous radius e u strain tensor re exumbrellar radius critical strain rf fibre radius E c circumferential stiffness rm subumbrellar radius E f fibre stiffness U radial coordinate within bell wall Ejm matrix stiffness (joint P sea water density mesoglea) E L stiffness parallel to fibres Syki compliance tensor Em matrix stiffness (bell CT stiffness mesoglea) E T transverse stiffness stress tensor f > resonant frequency effective circumferential stress F force t time F m driving (muscle) force T circumferential tension F s spring restoring force X wall thickness Y specific damping coefficient V subumbrellar cavity volume G T L bulk modulus, fibre plane V f volume fraction G T T bulk modulus, transverse vw wall volume plane h bell height w elastic energy hs shoulder height natural frequency k spring constant X circumference Acknowledgements Thank you to the people and organisations who helped make my thesis a reality. The staff and divers of the Bamfield Marine Science Centre, Coastal Ecosystems Research Foundation, and West Wind Sealab Supply collected specimens for me. Dr. Andy Spencer got me started videotaping the animals and helped me understand jellyfish wiring and muscle activation. The late Dr. Nikita Grigoriev was a big influence in the early days of my graduate career - much of the data presented in Chapter 2 owes its quality to his guidance. Dr. Bill Milsom got me thinking about flow-limited resonators, and lent me a pressure transducer for a couple of years (until I broke it!). Dr. Boye Ahlborn suggested the vortex model of added mass. I owe Tara Law a debt of thanks for needing a machine vision system and providing die impetus I needed to get to work on it. I'd also like to thank her for her hospitality during an extended stay in Bamfield. Nicola Cameron sat patiently in front of a video monitor for two weeks to create the biggest part of the dataset behind Figure 2.5. Eric Luiker introduced me to Lab View many years ago, sparing me counUess hours of painful Pascal and C++ coding - for that I will be forever grateful. Christian Neumer lent me a computer to do some modelling and analysis during a "holiday" in Europe one Christmas. Thank you to Drs. Robert Miura & Gary Sl\ajer (UBC) and Jim Hill & Michael West (U of Wollongong) for discussions on stress and strain in thin and thick walled cylinders. Tliank you also to the members of my supervisory committee, Drs. Gosline, Blake, Milsom, Spencer, and Ahlborn for comments on thesis drafts and for numerous discussions about jellyfish locomotion throughout my graduate career. Thank you to Dr. Margo Lillie, Tara Law, and the members of Dr. Gosline's research lab for comments on my work and on earlier drafts of this manuscript. This work was supported by research grants from the Natural Sciences and Engineering Research Council of Canada, the Zoology Department of the University of British Columbia, and die Coastal Ecosystems Research Foundation. Much of data collection for Uus paper was conducted while I was funded by a University Graduate Fellowship from the University of British Columbia. Thank you to my current employer, the University of Wollongong, Australia, for providing me with a computer and time off to finish the write-up of this thesis. Finally, Uianks are due to my wife, Susanne, and family for their unwavering support of my seemingly endless late nights in the lab & office (yes, Liam, we can go play now!). was happy quite h, on a whim, ich comes first, thrust or fill? d the jelly such a spill ere writhing in the brill ow to swim... -WM, Vancouver, 1996 1 Chapter 1. General Introduction Jellyfish Locomotion This thesis is about the energetics of jellyfish locomotion, specifically some of the morphological and behavioural adaptations that minimise the cost of locomotion. Jellyfish, like all animals, must allocate the energy they obtain from food to maintenance, repair, and activity - the remainder can be put towards growth. A reduction in energy costs will allow more investment in growth, wliich will be in some way related to the animal's overall genetic fitness. So for a given amount of energy intake, any energy savings ought to be favoured by natural selection. The concept of a gait lias long been recognised in terrestrial locomotion, but it is relatively new to aquatic locomotion (Drucker 1996). One can identify four gaits in the swimming of fish (from slowest to fastest): gill effluent jet propulsion, median & paired fin paddling, periodic undulatory propulsion using the body and caudal fin, and transient or fast-start swimming, again using the whole body, but a different set of muscles. Corresponding gaits have been described for squid: low-speed respiratory jetting, normal cruising head first, and escape jetting mantle first (Gosline & DeMont 1985). Different muscle groups and skeletal structures are used for the different gaits (Gosline et al. 1983, Gosline & Shadwick 1983a). One can similarly recognise a set of gaits in jellyfish locomotion. Mills (1981) and Arkett (1984) describe two gaits for Polyorchis penicillatus: sink-fishing and active swimming, while Donaldson et al. (1980) describe an additional escape gait in Aglantha digitate. Later in this work, I propose a further division of the active swimming gait into intermittent and resonant swimming. In intermittent swimming, the animal contracts and refills, then coasts for a while before contracting again. The jet frequency can vary from very occasional (almost sink-fishing, but directional, with tentacles retracted) to nearly resonant. In resonant swimming, the bell begins to contract as soon as it has completely refilled. Gladfelter (1972) and Arkett (1984, 1985) described these two gaits, and DeMont (1986) suggested that resonance in the latter could provide as much as 30% energy savings. In Chapter 5, using a non-linear model fit to experimental data, I will find a similar savings for mid-sized animals, and much higher savings potentially available to smaller animals. That an animal as simple as a jellyfish exhibits a variety of behavioural gait patterns is perhaps a bit ofa surprise. The hydrozoan jellyfish is one of the simplest of freely moving metazoans. The animal lias only one swimming muscle along the underside of its bell. When it contracts the muscle, it generates forward thrust either through jet propulsion, as in the case of Polyorchis and other prolate species, or in the case of oblate species, through a drag-based mechanism reminiscent of sculling (Colin & Costello 2002). In either case, to reopen the bell and refill the cavity, the jellyfish must rely on elastic energy stored in the deformation of the bell during the contraction. Gladfelter (1972) described the morphology of the bell of 2 Polyorchis, including a set of radially oriented fibres, which he concluded must be the element responsible for this energy storage. The fibres were studied in detail by Weber & Sclunid (1985). They were recently identified as microfibrils made of a protein homologous to mammalian fibrillin (Reber-Muller et al. 1995). DeMont (1986) measured die material properties of the mesoglea, then the elastic behaviour of the whole bell, concluding that it was indeed possible to store enough energy in the bell to account for the refilling phase. Gladfelter (1972) described the spring behaviour as non-linear and DeMont (1986) quantified it, and although both assumed that the non-linearity was due to die mesoglea first folding, then being stretched radially, neither modelled the system in sufficient detail to be able to confirm dieir assertion. It is clear tiiat die bell acts as a spring, but what role does it play in determining the behaviour of the jellyfish locomotory system? Certainly, a spring-mass system will oscillate, and, if left alone after a disturbance, at a frequency in some way directly proportional to the stiffness of the spring and inversely proportional to die mass of the system (French 1971). This frequency is referred to as the natural, or resonant, frequency. If there is significant friction in die system, dien die resonant frequency will be lower. If the oscillator is driven by an external force, such as die swimming muscle, tiien it can operate at other frequencies, but the energetic cost will be higher. DeMont & Gosline (1988c) modelled die jellyfish as a linear harmonic oscillator, and suggested that die stiffness of die jellyfish bell was such tiiat die animal might be making use of die energy savings associated witii resonance at its usual swimming frequency (1Hz). The success of their model was limited, however, by its reliance on linear mathematics, and on the assumption that die stress-strain beliaviour tiiey measured while die animal was glued to Plexiglas was representative of the elastic behaviour of the free-swimming animal. Anodier limitation of the linear oscillator model is diat the animal is restricted to a single operating frequency if it is to benefit from die resonant energy savings. Gladfelter's (1972) observations of large variation in swimming frequency, even within a single animal, seem to suggest that the animals only rarely "get it right," and spend most of their time off-resonance, away from die potentially huge energy savings. In Chapter 5,1 propose and show that the non-linear spring beliaviour described by Gladfelter (1972) and quantified by DeMont (1986) provides a mechanism for the animals to benefit from resonance at multiple frequencies. The Study Animal Ecology Polyorchis penicillatus (Figure 1.1), also known as P. montereyensis (Rees & Larson 1980), is a small (~1-5 cm long) hydrozoan jellyfish, found along the Pacific coast of North America from southern California to the Queen Charlotte Islands (Arai & Brinckmann-Voss 1980) and soutiieast Alaska (Rees & Larson 1980). Aldiough specimens have been found in most parts of the species' range in all months of the year, the 3 Figure 1.1. The penicillate jellyfish, Polyorchis penicillatus Eschcholz. 4 animals are most abundant in the summer in the inlets of the west coast of Vancouver Island (Arai & Brinckmann-Voss 1980). Large numbers of Polyorchis have also been seen in the waters of Queen Charlotte Strait, between northern Vancouver Island and the British Columbia mainland (Brinckmann-Voss 1977, Megill 1996, Stelle 2001, Stelle & Megill, in prep.). There is one published report of the hydroid colony, from a specimen found in Departure Bay, near Nanaimo (Brinckmann-Voss 1977), but it is now known that the specimen in question was in fact Sarsia bella (Brinckmann-Voss 2000). With the retraction of that one report, there has to date been no published description of the hydroid of any of the Polyorchis species (Brinckmann-Voss 2000). Previous workers (Gladfelter 1972, Arai & Brinckmann-Voss 1980, Mills 1981, Arkett 1984, DeMont & Gosline 1988a) reported often finding the medusae at the heads of inlets, in sea-grass (Zostera) beds, and over muddy bottom substrate (Mills 1981, Arkett 1984). Since beginning this study, I liave found them to be equally, if not more, abundant, on the outer coast, in exposed sandy-bottom bays (Stelle & Megill, in prep.). Similar observations have been made by other workers in the Goose Group, near Bella Bella, BC (Eminett et al. 1995). Polyorchis feeds both in the water column and on the bottom (Mills 1981, Arkett 1984). In the water column, they are known to feed on cumaceans and gammarid amphipods, among others (Arkett 1984), while on the bottom, they feed on benthic and demersal organisms (Arkett 1984). Recent observations of Polyorchis on the Central Coast of British Columbia add the mysid, Holmsimysis sculpta, to the list of prey items (Stelle 2001, Stelle & Megill, in prep.). Animals used in the experiments reported here were collected from several locations along the British Columbia coast, including Bamfield Inlet & Pachena Bay near Bamfield, Esquimalt Harbour, near Victoria, and Hardy Bay, Port Hardy. Animals were kept in running seawater aquaria at the Bamfield Marine Sciences Centre, or in a recirculating seawater aquarium at the University of British Columbia, until use. They were fed YneArtemia every other day, and only those animals which remained healthy (i.e. actively swimming and transparent), were used for the experiments reported here. Physiology Mesoglea The hydrostatic skeleton of Polyorchis is made up of three skins and two types of mesoglea. The outer layer of mesoglea is contained between the exumbrellar epithelium and the gastrodermal lamella. Quite stiff, it is made of a mucopolysaccharide matrix, reinforced by elastic fibres (Bouillon & Vandermeerssche 1957, Bouillon & Coppois 1977, Gladfelter 1972, Weber & Schmid 1985). There are two types of fibres in the mesoglea of Polyorchis: thick (1.5-1.8 um: Gladfelter 1972, Weber & Schmid 1985, DeMont 1986), radially-oriented fibres (Figure 1.2) made of a protein homologous to mammalian fibrillin (Reber-Muller et 5 P Figure 1.2. Polyorchis morphology. Note radially oriented fibres (RF) and joint mesoglea (JM). Top panel, longitudinal section. Bottom panels, cross section. P, peduncle; RC, radial canal; G, gonad; M, manubrium; BM, bell mesoglea; V, velum; CM, circumferential muscle; RM, radial muscle; EU, exumbrellar epidielium; SU, subumbrellar epidielium; GL, gastrodermal lamella; AR, adradius, IR, inter-radius; PR, perradius; XS, cross-section. Redrawn from Gladfelter (1972). 6 al. 1995), and a thin (6-15 nm) fibre network made of collagen (Chapman 1953, Bouillon & Vandenneerssche 1956, Weber & Schmid 1985). The inner layer of mesoglea, located between die gastrodermal lamella and the subumbrellar epithelium, is less stiff, lacking die radially-oriented elastic reinforcing fibres, but is still a mucopolysaccharide gel. There are fibres present (Weber & Schmid 1985), but tiiese are sparsely distributed and arranged randomly. This so-called joint mesoglea is divided into eight regions by the longitudinal interconnection of the subumbrella and gastrodermal lamella (Spencer 1979). The regions are triangularly shaped in cross-section (Figure 1.2), and their lower stiffness allows the bell mesoglea to fold around them during deflation (Gladfelter 1972, Weber & Schmid 1985, DeMont 1986). Gladfelter (1972) analysed the geometry of the folding (Figure 1.3), comparing die jellyfish bell to a cylinder witiiout joints. He found that when die muscle shortened by 44%, die subumbrellar surface of the mesoglea was only compressed by 14%. Likewise strain in the exumbrellar surface was 19% less tiian the 21% predicted witiiout the joints. Radial strains were also less: 34/36/16% at die inter-, per-, and ad-radii, respectively, compared to 46% uniformly for die unjointed model. The significance of this strain relief is that more of the force generated by the muscle can go into expelling water, rather tlian into deforming the bell (Gladfelter 1972). As the differing radial strains show, the deformation is not evenly distributed throughout the bell. Gladfelter (1972) showed tiiat the observed strains were inversely proportional to the local density of radial elastic fibres. DeMont (1986) quantified die elastic behaviour of the whole bell structure in deflation (Figure 1.4). The non-linearity of the stress-strain curve is a reflection of the geometry of the structure. The pressure-volume, and hence stress-strain, behaviour of any cylinder is non-linear (Young 1989). The joints increase die non-linearity by allowing the bell to fold before being stretched radially. As die hinges reach their maximum flexure, the contraction begins to stretch the radial fibres, since die volume of the mesoglea remains constant and die length of die animal does not change. Fung (1984) stresses the importance of quantifying this non-linearity in order to make predictions about the behaviour of the system. When the swimming muscles relax at the end of a contraction, die energy stored in die elasticity of the bell causes it to re-extend, first by relaxing die radial elastic fibres, then by unfolding. When die bell reaches its resting circumference, it still has inertia, and hence continues to extend, stretching the mesogleal matrix as well as die sub- and ex-umbrellar epitiielia. The behaviour of the bell spring in extension was not investigated by previous authors (Gladfelter 1972, DeMont 1986), but it seems reasonable to predict tiiat the stiffness of the bell in extension should rise quickly as die epitiielia are stretched circumferentially. Musculature Unlike squid, which have an antagonistic pair of swimming muscle systems acting on die hydrostatic skeleton (Packard & Trueman 1974, Wainwright et al. 1976, Gosline et al. 1983), jellyfish liave only one set of swimming muscles, which are antagonized by the elasticity of the bell, as described above (Romanes 7 Figure 1.3. Folding behaviour of Polyorchis mesogleal bell, in cross-section, midbell. A, relaxed; B, contracted; C, hypothetical unjointed jellyfish. The stiff bell mesoglea folds around the compliant joint mesoglea, providing significant strain relief compared to a cylinder without joints. Redrawn from Gladfelter (1972). 8 Elastic Behaviour of Mounted Jellyfish Preparation o -50 -I 1 1 1 1 -0.4 -0.3 -0.2 -0.1 0.0 Circumferential strain Figure 1.4. Stress-strain behaviour of Polyorchis mesogleal bell in deflation. Redrawn from DeMont & Gosline (1988a). Low initial slope reflects the folding of the bell around die mesogleal joints. The stiffness increases widi higher strain due to the deformation of the stiffer bell mesoglea and its radially oriented reinforcing fibres. The curve is DeMont's (1986) quadratic fit to the data. 9 1876, Gladfelter 1972, Daniel 1983, DeMont 1986). The swimming muscle is striated, and consists of a layer of modified epithelial cells on the subumbrella. The cell bodies are close-packed to form an epithelium, and the myofilaments are radially distant from the cell bodies (Mackie & Passano 1968). In scyphozoan medusae, the myofilaments run both circumferentially and longitudinally (Gladfelter 1973), but in hydrozoa, they run circumferentially only. In Polyorchis, the subumbrellar muscle sheet is only one cell thick (Gladfelter 1972, Singla 1978b, Satterlie & Spencer 1983, Spencer 1995, Lin & Spencer 2001), but in other species, there can be extra folding of the muscle sheet to increase the cross-sectional area (Gladfelter 1973). Gladfelter (1972) describes the subumbrellar muscle sheet of Polyorchis as divided into four quadrants, separated by the radial canals. Spencer (1979) describes it as a single sheet, contiguous past the radial canals. The muscles are innervated from the base by the inner nerve ring, and from the "edges" by radial motor neurons running along the radial canals (Spencer 1978). The duration of the motor action potential around the edge of the muscle sheet is adjusted as the signal travels so that the muscle action potential is initiated everywhere simultaneously (Spencer 1982). The motor neurons around the edges of the sheet synapse with the epithelial cells to create muscle action potentials, which are transmitted in myoid fashion across the muscle sheet (Spencer 1978, 1982). The musculo-epithelial cells are arranged like bricks in a wall, longer circumferentially than longitudinally. The muscle action potential therefore lias to negotiate fewer gap junctions and hence can travel faster in the circumferential direction (Spencer 1995). The result is staggered muscle depolarisation, as shown in Figure 1.5 (Spencer 1982). This staggering in the circumferential direction means that the muscle first contracts near the joints, reinforcing the tendency of the mesoglea to fold there, then later contracts between the joints, extending the radial fibres. The other set of muscles involved are those of the velum. By contracting during the jet stroke, they create a nozzle, increasing the velocity of the flow out of the bell. This increases the total forward thrust, as evidenced by Gladfelter's (1972) velum removal experiments. The velum also plays a role in turning. The radial muscles of the velum contract asymmetrically, pointing the nozzle off-axis, causing the animal to turn (Gladfelter 1972, Singla 1978b, Spencer 1979). Three other sets of muscles exist, though none directly involved in swimming. There are longitudinal muscles in the tentacles, which contract to haul the tentacles in when prey is caught or at the beginning of a swimming bout (Spencer & Schwab 1982). Radial muscles lie along the radial canals, and serve only in the "crumpling" response (Spencer & Schwab 1982, King & Spencer 1981) to excessive stimulation (presumably a defense mechanism that gets the delicate nerve rings around the margin of the bell inside the bell cavity, and hence out of harm's way). And finally there are endodermal muscles associated with the manubrium and digestive system (Gladfelter 1972). 10 Figure 1.5. Contraction timing of Polyorchis swimming muscle contraction. The muscle action potential is initiated simultaneously at all sites around die periphery of the muscle sheet by motor neurones running parallel to the radial canals, and travels across the sheet in myoid fashion. The action potential travels faster circumferentially tlian longitudinally - hence die staggered activation. The earlier contraction at the apex of the sheet ensures that thrust is produced efficiently. The earlier contraction at the edges of the sheet reinforces die tendency of the mesogleal bell to fold there. (Redrawn from Spencer 1982). Shaded area of inset shows location of muscle quadrant. 11 Structure of Thesis The remainder of this thesis is divided into five chapters (Figure 1.6). The next chapter is a study of the kinematics of jellyfish swimming, which describes a set of four gaits used by die animals, including one in which the animal may be resonating. The tiiird chapter looks at die origins of elasticity in die jellyfish mesoglea, beginning with a measurement of the material properties of the mesoglea and the radial fibres distributed throughout it. The fourth chapter develops an anisotropic thick-walled cylinder model to predict the non-linear elastic beliaviour of the bell, tiien verifies die model against experimental data. The fifth chapter considers the implications of the non-linear stress-strain behaviour on the oscillating system, concluding that the system does in fact resonate at die frequency predicted by die model, and that the resonant frequency is proportional in some way to die amplitude of the driving force. It also concludes that the mounted preparation resonates at a frequency much greater than the swimming frequency of Polyorchis. Modifications are made to the tiuck-walled cylinder model developed in Chapter 4 to account for the effects of tethering necessary to measure the stress-strain behaviour of the jellyfish bell. A new vortex model of the system mass is proposed. The mass and cylinder models are then used to predict die behaviour of the untethered, free-swimming animal. Spring stiffness, mass and damping are scaled witii bell size to allow a prediction of resonant frequency to be made for jellyfish of all sizes. The prediction is tiien compared to die kinematics of free-swimming animals to present a new understanding of the mechanics behind jellyfish locomotion. The final cliapter ties the thesis together and discusses its results in the context of other research. 12 Kinematics I Resonant gait i Non-linear harmonic oscillator Chapter 2 • Origins of non-linear elasticity Material properties Mesoglea Fibres Geometry Chapter 3 Transversely isotropic thick-walled cylinder model I Quasi-static predictions & experimental verification for mounted jellyfish preparation I Dynamic predictions & experimental verification for mounted jellyfish preparation Chapter 4 Free-swimmer Change of boundary conditions Chapter 5 Fluid dynamics Mass Prediction • Verification by comparison with kinematics Figure 1.6. Flowchart of thesis structure. 13 Chapter 2. Kinematics, Gaits, and Resonance. Introduction Animals use a variety of behaviours to take advantage of energy storage and exchange mechanisms to minimise the cost of transport (energy required to move a given mass a given distance). In this chapter, I will focus on gait patterns, particularly on the role of elastic energy storage, damping, and resonance in the active swimming gaits ofa hydromedusan jellyfish, Polyorchis penicillatus. In terrestrial locomotion, since air resistance is negligible, (except at very high speeds: Alexander 1976), legged animals have only to overcome the deceleration/acceleration the body undergoes with each footfall. Low speed walking exchanges vertical potential energy for horizontal kinetic energy (Cavagna & Margaria 1966, Cavagna 1969, 1975, Alexander 1984). This could be thought of as storing horizontal kinetic energy in the vertical displacement of the centre of mass. Higher speed gaits use the elasticity of tendons and ligaments to store energy during landing, which can then be used during takeoff (Alexander 1974, 1977, 1984, Alexander & Vernon 1975). If cost of transport is plotted against speed for terrestrial animals, each gait will appear as a u-shaped curve (Taylor 1977). Animals prefer to use the speeds where cost of transport is minimised, namely at the minimum of the curve for each gait, and have to be specially trained to move at other speeds. In aquatic locomotion the opposing force is hydrodynamic drag, which at high speeds is proportional to the square of velocity (Batchelor 1967), causing the cost of transport to rise quadratically with velocity. Aquatic animals have also been shown to use elastic energy storage mechanisms: fish and dolplun skin fibres (Alexander 1987, Pabst 1996, Cheng & Blickhan 1994), cetacean tendons (Blickhan & Cheng 1994), squid elastic fibres (Gosline & Shadwick 1983a,b), scallop hinges (DeMont 1990), and jellyfish mesoglea (DeMont & Gosline 1988a). McHenry et al. (1995) discuss how the tapering geometry of a fish's body may also store energy. An analogue of the low speed walking energy storage and exchange may exist in the interaction of the animal with the fluid. Ahlborn et al. (1991) showed that a fish could destroy the start-up eddy created by the sideways deflection of its tail to get an extra boost forward. Vogel (1985) and Cheng et al. (1996) show that the scallop may get help reopening its shell from the lift created by its motion through the fluid. One can speak of a set of gaits in aquatic locomotion as well (Webb & Gerstner 2000). At very low speeds, fish may use the jet propulsion of their gill effluent to move around. At low to moderate speeds, most fish paddle to some extent using median and paired fins (MPF). At higher speeds, the whole body is undulated using mostly red (aerobic) muscle. At very high speeds, the fish also uses undulatory propulsion, but its white (glycolytic) muscles are recruited. The two higher speed gaits are referred to as body and caudal fin 14 swimming (BCF-periodic and BCF-transient, respectively). A number of authors have worked on the mechanics of each gait (paddling: Blake 1981; BCF-p: Webb 1971a, b; BCF-t: Webb 1978, 1983, Wardle et al. 1995). Specialists have evolved among die fishes to take full advantage of each of Uiese gaits (Webb 1984). The same variety of gaits also exists in jet-propelled squid (O'Dor 1988). At rest, respiration requires a continuous flow of water in and out of the mantle cavity, creating a small jet. At low speeds, die animals use a pair of fins along die body. These fins are folded in against die body at higher velocities, and die animal uses jet propulsion to propel itself head forward by directing its nozzle backwards. At higher speeds, the animals swim backwards, head trailing. Maximal escape velocities are attained by hyper-inflating (Gosline & DeMont 1985). Squid also possess analogues of fish red and white muscle, which are recruited sequentially as required to generated the desired level of thrust (Mommsen et al. 1981). The cost of transport does not seem to explain changes in gait for fish. This is likely due to die change in gait having no fundamental change in the mechanics of locomotion - the two higher-speed gaits (BCF-p and BCF-t) are hydrodynamically identical, die only change being which muscle fibres are recruited -unlike the gait transitions in terrestrial animals, where different energy storage systems are employed (gravity and inverted pendulum, leg springs, torso springs). Unlike the continual thrust tiiat can be produced by a rotating propeller, most aquatic animals must use intermittent propulsion in the form of body waves (BCF gaits), appendages (e.g. paddling), or jet propulsion. The body wave produces tiirust almost continuously, but the other two swimming modes are intermittent. Appendages can be folded or feathered during die return stroke, in order to minimise drag and hence die cost to forward locomotion performance. Jet propulsion, on die other hand, requires a refill stroke wherein surrounding fluid must be sucked back into the body cavity. If the aperture through which the cavity is refilled is located at die trailing end of the animal, such as in jellyfish, substantial reverse thrust will be created. Squid and salps, which refill tiirough apertures at the leading edge of their bodies, may be able to gain a small amount of additional forward tiirust tiirough suction, though this does not appear to have been investigated to date. Daniel (1983) investigated the added cost of locomotion for a jellyfish due to the intermittent acceleration and deceleration of the fluid around it. Daniel (1984) shows that once squid are at high speed, die acceleration reaction due to die entrained mass of water becomes relatively unimportant compared to the drag. This explains why the cost of transport for fast-moving intermittently propelled squid is similar to that of fish which are able to produce almost continuous tiirust (O'Dor 1988). Jellyfish, however, move at slow speeds (Gladfelter 1973), and must continuously overcome die acceleration reaction problem (Daniel 1984). One would expect tiierefore tiiat their cost of transport should be higher compared to similarly sized fish, but tiiis does not appear to be die case. Larson (1987) suggests tiiat it is die low specific metabolic rate of jellyfish tiiat keeps die cost low. 15 Jellyfish also use a number of gaits. Singla (1978a) and Donaldson et al. (1980) describe two gaits for Aglantha digitate: a regular, rhythmic gait, and an escape response. Mills (1981) and Arkett (1984) describe two gaits for Polyorchis penicillatus: sink-fishing and active swimming. In sink-fishing, the negatively buoyant animal sinks slowly through the water column, outstretched tentacles first. The tentacles greatly increase the drag on the falling animal, but in so doing, accelerate more slowly than the animal. Just before the tentacles become tangled, the animal contracts once to stop its descent and reposition its body relative to the tentacles. Just prior to the contraction of the swimming muscle, there is a contraction in the longitudinal muscles within the tentacles, which straightens them out (Spencer 1978, Spencer & Arkett 1984). The animal then begins to sink again, relaxing and re-extending the tentacles to reset its fishing net. In the present document, I divide previous authors' (Mills 1981, Arkett 1984) active swimming gait into transient and resonant swimming. In transient swimming, the animal contracts and refills', then coasts for a while before pulsing again. Tentacles are retracted to some degree by longitudinal muscles which are activated by nerve stimuli just prior to the contraction of the swimming muscles (Spencer 1978, Spencer & Arkett 1984). The pulse frequency can vary from very occasional (almost sink-fishing, but directional) to nearly resonant. In resonant swimming, the bell begins to contract as soon as it has refilled completely. The high frequency of muscle contractions ensures tliat the tentacles are completely retracted in this gait. The two modes of swimming are mechanically identical, the only difference is in the presence or absence of the quiescent period between pulses. The jellyfish has only one set of swimming muscles, and these can only contract the bell. The refilling stage is purely passive, powered by elastic energy stored in the deformation of the bell toward the end of the jet phase (DeMont 1986; Gladfelter 1972). The animal is therefore constrained to operate at its natural frequency during the refilling (recoil) stage, no matter which gait it selects. DeMont & Gosline (1988c) modelled the system as a linear harmonic oscillator. I will show tliat the linear model works to some extent, but requires adjustment to correctly predict the observed behaviour of the free-swimming jellyfish, particularly with regards to the non-linearity of the mesogleal spring. Materials & Methods Live penicillate jellyfish, Polyorchis penicillatus (Figure 2.1), were obtained by SCUBA divers from Bamfield Inlet and Pachena Bay, off southern Vancouver Island, British Columbia. They were kept at the Bamfield Marine Sciences Centre in running sea water aquaria and fed live brine shrimp, Artemia sp., daily until use. The behaviour of freely swimming jellyfish was recorded on videotape using a Sony Hi-8 1 This two-part behaviour of muscle contraction and passive elastic refilling is described throughout the remainder of this thesis as a pulse. 16 Figure 2.1. Morphological measurements (photograph by the author, drawing modified from Gladfelter 1972). h, bell height; hs, shoulder height; r0, resting bell radius; z, wall thickness. Mean h, = 0.77h (+/-0.02h, n=44), mean r0 = 0.3 lh (+/- 0.04h, n=44), mean x = 0.12h (+/- 0.02h, n=17). The peduncle is the region at the base of the manubrium, while the apex is defined here to be the region above the shoulder. 17 camcorder. Three filming arenas were used in tiiis study. The first consisted of a large circular tank (2m diameter), approximately 30cm deep, with a mound of rocks covering a standpipe in the centre. The second was a 25 gallon aquarium with a black plastic sheet taped to the back of it. The third arena was a 10cm x 10cm x 40cm Plexiglas tank. A section of black cardboard was taped to the back of the tank. In the large circular tank, a gende current (roughly 3 m/min) was created by a nozzle at the outer edge of the tank, while in the other two arenas, zero flow was maintained while filming. No special illumination was provided for the large tank - the animals were simply filmed in die ambient (low-level) light as they drifted or swam past the camera, which was mounted above a section of the tank. The camera's field of view was adjusted so that it covered die tank from die central standpipe to the outer wall, and area of about 0.5 m2. From die four hours of recordings made, forty 10-second segments were selected at random intervals of between 3 and 10 minutes. One frame of each segment was digitised using a video capture card (Snappy 3.0, Play Incorporated, Jersey, UK), and printed. All of die jellyfish visible on the printout were labeled, and their behaviour tabulated by tracking it during the ten seconds following the printed frame. Behaviours were classed as drifting (no pulses during the ten seconds), transient (pause between pulses), resonant (no pause between pulses), otiier swimming (ineffective contractions, bizarre swimming behaviours), or non-swimming. Though likely not exactly representative of the natural behaviour of the animals (e.g. Arkett 1984, Mills 1981), die observation is included here to demonstrate tiiat the active swimming gaits are an important part of the overall behavioural budget of the animals. The smaller tanks were illuminated from die side (perpendicular to the axis of die camera) using a manual slide projector. Animals were filmed as they swam freely around die aquarium. A tripod-mounted Hi-8 video camera was focused on a single animal in the tank and followed it around such tiiat the animal always filled as much of the frame as possible. To videotape animals in the small Plexiglas tank, the camera was fixed in place, and turned on its side so tiiat the long edge of die frame was parallel to die long dimension of the tank. This made it possible to record additional pulses from the animals as they swam vertically upwards in the narrow tank. Video sequences in which die animals strayed too close to die edges of the tanks, or where they left the illuminated part of the tank, were discarded. Ten minutes of clear sequences (avg. duration, 6 sec. @ 30Hz) were chosen from die 8 hours of recordings made, then copied to digital tape via S-Video using a Canon ZR10 camcorder. The digital recordings were transferred via FireWire to a Macintosh computer using iMovie (Apple Computer, Cupertino, CA) and converted to bitmap sequences for analysis using QuickTime Pro (Apple Computer). Digitisation was done using custom software written in die Lab View programming language (National Instruments Corp, Austin, TX). Depending on die quality of the image, the outline of the jellyfish was 18 traced either using an automated edge detection algorithm, or by hand using a mouse. The former produced much better results and so was used whenever possible. The outlines were then rotated to a vertical orientation, the tentacles were removed, and the remaining outline of the bell was cut at the level of the shoulder, discarding the apex. The average outer diameter of the jellyfish was calculated as a function of time from the distance between the resulting two lines in each frame. Results Morphology Measurements were made as defined in Figure 2.1 of the bell height (h), shoulder height (h5), and margin radius (r0) of forty four animals. Data for an additional twelve animals were extracted from Gladfelter (1972). Measurements were also made of the wall tluckness (t) of 17 jellyfish. There was no correlation of margin radius, shoulder height or wall thickness to bell height for the animal size range studied in this paper (Figure 2.2). There was only one individual larger than 30mm included in the dataset, so it is possible, and anecdotal observations suggest (A.N. Spencer, pers. comm.), tliat a correlation would be observed if larger animals were included. However, for the size range studied in this paper [7mm < h < 30mm], the animals can be assumed to scale geometrically. The mean shoulder height ratio for the animals studied in this paper was 0.77h (+/- 0.02h, n=44), and the mean inner margin radius was 0.28h (+/- 0.04h, n=56). Mean wall tluckness was 0.12h (+/- 0.02h, n=17). Gaits Four gaits can be described for Polyorchis: hopping, sink-fishing, transient and resonant swimming2. The first two gaits are used in fishing, with tentacles extended, while the remaining two are used by the animals to cover distances (vertically: Mills 1981, and horizontally: Arkett 1984), characterized by the retraction of the tentacles, particularly in the resonant gait. Fishing gaits In hopping, the animal stands on its tentacles on the bottom and occasionally contracts briefly to stir up the mud, and sometimes move a small distance laterally (Mills 1981). The animals also sometimes use this gait to re-orient themselves to a vertical position (Mills 1981). Sink-fishing (Mills 1981, Arkett 1984) describes a behaviour in which the animal has its tentacles fully extended and is slowly sinking due to its negative buoyancy, then occasionally jets to slow its descent, maintain its position in the water column, or even sometimes climb slowly. The bell sinks slightly faster than the tentacles, so the behaviour may be a means of keeping the tentacles from tangling, or to maintain 2 A movie is available on the internet at: http://www.zoology.ubc.ca/~megill/pubs/jfish/moviel.html 19 Jellyfish Fineness Ratios 0.5 0.5 10 15 • Fineness ratio m. Thickness ratio 20 25 30 Bell height (mm) Figure 2.2. Polyorchis morphometries as a function of size. The bell height, shoulder height and margin radius of forty-four animals were measured during die course of the experiments reported in tiiis paper, and data for twelve additional animals were reported by Gladfelter (1972). There is no correlation of fineness ratio on bell height (R2 = 0.015). A similar lack of correlation was found between shoulder height and bell height. Wall thickness was measured for 17 animals, and no correlation with bell height was found. No data was collected in this study for animals larger tiian 30mm bell height - the 40mm individual was described by Gladfelter (1972) - so it is possible, and anecdotal observations suggest, that a correlation might be observed if larger animals were included in the data set. However, for die size range (7-30mm) studied in tiiis paper, die animals can be considered to scale geometrically. 20 Jellyfish Kinematics - Sink-fishing 1.0 2.0 B 1.5 Time (s) Jellyfish Kinematics - Transient (Fast) Time (s) Jellyfish Kinematics - Transient (Slow) Figure 2.3. Damped oscillations in sink-fishing (A) and transient swimming (B,C). Normalised amplitude is the instantaneous diameter divided by the resting diameter. The bold horizontal line is the normalised resting diameter. Note the overshoot and subsequent damped oscillations. In Trace B, the animal is beginning its next contraction just as the second damped oscillation reaches its maximum overshoot. This behaviour was not typical. Trace C is a more typical oscillation in which the duration of the quiescent stage was not predictable. Sampling rate (all 3 traces): 30Hz. Jellyfish bell heights: Trace A (Jellyfish #10) 24.9mm, B (Jellyfish #9) 17.0mm, C (Jellyfish #12) 22.3mm. Time (s) 21 an efficient predatory net. A typical trace is shown in Figure 2.3A. Swimming gaits Transient swimming In transient swimming, die animals pulse the bell at irregular intervals, coasting with the bell relaxed for a certain distance before pulsing again. The pulses tiiemselves are of a characteristic shape and duration, but the coast duration is irregular. In this gait, damped radial oscillations are often detectable after the refilling stroke of the pulse. Typical traces are shown in Figure 2.3, panels B & C. This gait seems to be used for travelling (as opposed to fishing), though the animals often do not completely retract their tentacles as they swim. Resonant swimming In die resonant gait, die animals begin their next pulse just as the bell reaches its maximum extension after refilling. The maximum extension is often larger tiian the resting diameter, providing die animal with an extra volume of water for die subsequent jet (similar to die hyperinflation in squid escape swimming reported by Gosline & DeMont 1985). This last gait is die most rapid, and one die animals use to cover distance (vertically: Mills 1981, and horizontally: Arkett 1984). The complete retraction of die tentacles in tiiis gait precludes effective fishing, though occasional prey items do get caught by die retracted tentacles (pers. obs.). Typical traces of animals starting from a stationary position are shown in Figure 2.4. Note the larger oscillation amplitude in die second and subsequent pulses. The hyperinflation is typically of the order of about 3-5% radial strain beyond die resting radius, which corresponds to an increased stroke volume of 9-16%. Since tiirust production in tiiis jet-propelled system is proportional to the mass of water ejected during the contraction, tiiis additional volume can be used by the jellyfish eidier to increase its forward acceleration, or to reduce die cost of generating die same acceleration. Interestingly, the duration of the pulses in Figure 2.4 follows the pattern predicted by Spencer& Satterlie (1981), with the second pulse longer tiian the first. Third and subsequent pulses were always shorter dian the first pulse. My sample size is however far too low, and die variability far too large to draw significant conclusions - what would be required is a study similar to Arkett's (1985) "treadmill" study with high-speed video. Relative importance of the gaits The relative importance of the swimming gaits in the behavioural budget of captive Polyorchis is shown in Table 2.1. The table shows the average proportion of animals in three of the gaits in forty ten-second intervals. On average, 20 animals were visible in each segment. Two thirds of the animals observed to be actively swimming were operating in the resonant gait, wliile the remaining third were swimming in a transient fashion. Most of the animals classified as drifting will in fact have been sink-fishing, although because of the short observation time, pulses were rarely observed. Arkett (1984) reported tiiat in the field, individual Polyorchis spend 90-95% of their time in the sink-fishing gait. He also reported tiiat thirty 22 Gait Proportion (%) Drifting 47 Transient 15 Resonant 31 Other 7 Table 2.1. Relative importance of jellyfish gaits. Shown are the average proportions of animals swimming in each gait in 40 ten-second video segments. Many of the animals scored as drifting were likely sink-fishing, but at interpulse intervals longer than 10s. 23 Jellyfish Kinematics - Resonant Time (s) B Jellyfish Kinematics - Resonant Figure 2.4. Typical traces of the resonant gait. Both animals started from rest at time t = Os. Normalised amplitude is the instantaneous diameter divided by the resting diameter. The bold horizontal line is the normalised resting diameter. Note the small overshoot and consequent increase in ejected volume in pulses subsequent to the first. Trace A (Jellyfish #13, h=13.3mm) was sampled at 30Hz and Trace B (Jellyfish #3, h= 16.9mm) at 12.5Hz. 24 percent of the individuals he observed swam actively in one to four extended bouts every ten minutes. Arkett (1985) describes captive animals as swimming more often, and for longer periods, than wild ones. Kinematics Swimming frequencies Average swimming frequencies were determined from video tapes of the animals swimming freely in a large aquarium (Figure 2.5). Average frequency for animals swimming in the transient or resonant gaits was defined as the number of pulses in a bout divided by the duration of the bout. In die resonant gait, swimming frequency ranged from 0.8 to 1.4 Hz, with an obvious decreasing trend of frequency with bell height, an observation in keeping with previous authors (Gladfelter 1972, Arkett 1985). In the transient gait, the swimming frequency was much lower, ranging from 0.2 to 0.8 Hz, and no trend was observed with bell height. That die resonant frequency of the system decreases with size is to be expected, as the mass of die system increases with size (see Chapter 5). The absence of a trend in the transient gait is a result of the arbitrary lengtii of the quiescent period between pulses in that gait. Arkett (1984) presented a similar dataset, and found a weak (r=0.323) decreasing relationship between swimming frequency and bell height. Pulse shape Individual pulses could be divided into two or tiiree phases of varying frequency. The first was a contraction phase, during which die bell diameter decreased by 10 - 20%. This was followed by the recoil phase, during wluch the bell refilled witii water in preparation for the next stroke, often overshooting the resting diameter. A tiiird phase was observed in sink-fishing and transient swimming, during which the diameter continued to oscillate at much reduced amplitude for one or more cycles (Figure 2.3). The pulses were generally fairly similar in shape, regardless of whether they occurred in transient or resonant gaits. The animals took on average 43% longer to recoil than they did to contract, an observation similar to the one made by Gladfelter (1972). Figure 2.6 shows the relative variability of the duration of the jet and refill strokes: standard deviations of the duration of each half cycle are plotted against each other for fifteen jellyfish. The variability of contraction frequencies was larger than that of the refill frequencies for all but two individuals, reflecting the difference between die active nervous control of die muscle-powered contraction (die "mood" of the jellyfish, sensu Passano et al. 1967), and the purely passive control (or lack thereof) of the spring-powered refill stage. Reynolds numbers The dimensionless Reynolds number is used in fluid dynamics to describe the relative importance of inertial and viscous forces in die fluid. At low Reynolds numbers, die drag on an object moving through the fluid is directly proportional to its relative velocity, while at high Reynolds numbers, drag is proportional to the square of velocity. The Reynolds number is defined as the product of a representative length, 1, and velocity, v, divided by the kinematic viscosity of the fluid, v (Batchelor 1967): 25 Jellyfish Swimming Frequencies in Active Gaits 1.6 1.4 -1.2 -£ 1.0 4 >^  o s 0 8 I" 0.6 H 0.4 -0.2 -0.0 10 # Resonant • Non-resonant — i — 15 20 25 Bell height (mm) I 30 35 Figure 2.5. Average swimming frequencies of jellyfish in the resonant and transient gaits. In the resonant gait, the swimming frequency is a decreasing function of size, since the effective mass of the oscillator is increasing, while in the transient gait, the swimming frequency is independent of size, reflecting the arbitrary length of the quiescent period between pulses. 26 Pulse Shape Variability 0.8 0.6 H 0.4 £ 0.2 4 0.0 0.0 0.2 0.4 0.6 Standard Deviation (Jet Frequency) 0.8 Figure 2.6. Pulse shape variability. Plotted are the standard deviations of the duration of each half of the swimming pulse for 15 individual jellyfish For all but two animals, the variation in jet frequency was much larger than the variation in refill frequency, reflecting the difference in control of the two halves of the swimming pulse. Refill frequency is controlled by the purely passive process of storing and releasing energy in the mesogleal spring, while jet frequency is controlled by the nervous system of the animal, and therefore dependent on the "mood" of the jellyfish. 27 R = — (2.1) V Tlie representative velocity here is the radial velocity of the margin (v = dr„ydt). The dimension that matters is the one which is perpendicular to die flow, which in diis case is the margin circumference (1 = 27tr0). The kinematic viscosity of seawater was taken from Batchelor (1967) as 10"6 s/m2. Reynolds numbers for the animals studied in tiiis chapter ranged from 75 to 250, and generally increased with bell height (Figure 2.7). These so-called intermediate Reynolds numbers suggest that the drag on the bell margin should be proportional to some power of the velocity between one and two. Note that at Reynolds numbers less than 50, the viscous forces in the fluid dominate the flow and inertial structures such as wakes and vortices do not persist. This observation will be important in the discussion of die mass of the jellyfish oscillator at the end of Chapter 5. Discussion Polyorchis is an active swimmer (Gladfelter 1972, Mills 1981, Arkett 1984, 1985, DeMont 1986). An efficient predator (Arkett 1984, Stelle & Megill in prep.), it spends a substantial amount of its time actively swimming from one place to another, presumably in search of patches of abundant prey (Arkett 1984). Once in place, it switches to passive fishing modes, either perched on the bottom, occasionally hopping to stir up the mud and benthic organisms, or extending its tentacles fully into a drift net, sink-fishing to maintain its position in the water column. Mills' (1981) active swimming gait can be split into transient and resonant modes. The former may be a foraging gait, used to cover significant distances, but often with tentacles at least partially extended, dragging its fishing net tiirough the water. The resonant gait is obviously used to cover larger distances, with bouts often continuing for indefinite periods, until an obstacle is encountered. Muscular control The mechanics of die stroke are the same in all gaits, yet both the jet and refill frequencies are highly variable, even for a single jellyfish (Figure 2.6). It is straightforward to understand how the jet frequency might vary, since the contraction is powered by a muscle under nervous control, but die variation of the refill frequency is more difficult to explain. In hydrozoan medusae, no muscle exists to power die return stroke, so strain energy stored in die deformation of the mesoglea must power the refilling of the bell cavity. Gladfelter (1972) noted tiiat since the frequency at which a stretched spring contracts is dependent on its material properties, which should be constant for any given jellyfish, die refill frequency should always be the same. He suggested that the swimming muscle might be remaining activated, and hence limit die speed at which die system can refill. However, Spencer & Satterlie's (1981) research on die timing of 28 Jellyfish Reynolds Numbers 10 15 20 Bell height (mm) 25 Figure 2.7. Reynolds numbers, referenced to the margin velocity and circumference. Each data point represents the average Reynolds number for all observed pulses made by a each individual jellyfish. The error bars are the standard deviations. 29 the muscle contraction suggests tliat this is not the case. They recorded action potentials and consequent force development in isometric preparations of jellyfish ranging in size from 6 to 45 mm (bell height). They show that the duration of the action potential is an increasing function of body size, and that the duration of the time to peak tension is an increasing function of the duration of the action potential. They state that the time for decay of tension stays relatively constant. From the strain gauge recordings they present, it would appear tliat the duration of the decay is approximately 200ms, and may possibly be shorter for smaller jellyfish (roughly 160ms). Indeed, Lin & Spencer (2001) report a decay duration of 130ms for jellyfish of 15-20inm diameter (approximately 20-33mm bell height, if converted according to the morphological relationships presented in Figure 2.1). Using this information, it is possible to derive a relationship between bell height and tire duration of muscle activation. Time to peak tension, duration of muscle activation and duration of jet phase are plotted together in Figure 2.8. The sliape of the muscle pulse, as presented by Spencer & Satterlie (1981) is reproduced on the left-hand vertical axis of the figure. In all cases, the peak tension was achieved long before the bell liad reached its minimum diameter. It is not possible to conclude from the figure tliat the muscle was not at all active during the refill phase, but as shown by the dashed line, it was producing very little force and still relaxing when the turnaround occurred. Note that Spencer & Satterlie's results were for isometric contractions, so the durations reported reflect an upper limit, since force development and decay generally take longer in isometric contractions than when the muscle is allowed to shorten (Josephson 1999). In any event, the important observation to make here is tliat, contrary to Gladfelter's (1972) prediction, the muscle is not actively resisting the refilling of the bell. Harmonic oscillator If the muscle is not the source of the recoil frequency variation, then what is? More detailed study of the mechanics of the oscillation provides the explanation. The force balance for the system can be written: m * ~ F m u s d e — ^spring ~ ^friction (2-2) If all of the terms are linear, and one considers the system during recoil, when F m m c i e is zero, this reduces to the damped harmonic oscillator equation, as proposed by DeMont & Gosline (1988c): m'x + bx + kx = 0 (2.3) where die dots denote differentiation with respect to time, m is the total mass of the system, b the damping coefficient, and k the spring constant. The solution to the equation is given by: -bl/ x(t) = Be / 2 m (cos(<y0 + sin(ft*)) (2.4) 30 Muscle Activation and Contraction Timing during Jet Phase Figure 2.8. Muscle activation and contraction dining during the jet phase. Solid triangles are durations of jet phase of all contractions reported here. Doited line is die predicted time to peak tension (tmax) as derived from Spencer & Satterlie (1981). Solid line is the predicted total muscle activation time (t,*), as derived from Spencer & Satterlie (1981). Dashed line is die duration of the pulse past peak to 30% of maximum (t30). Assuming tiiat the bell begins to move as soon as the force begins to rise, tiien the muscle has always reached max tension and is relaxing before die end of the jet phase. To the left of the y-axis is the muscle pulse shape as described by Spencer & Satterlie (1981), Figure 5, on which die muscle contraction timing model was based. 3 1 where x represents the instantaneous circumference of the animal, t is time, B an arbitrary constant, and: where f = co / 2n is termed die natural frequency3. DeMont & Gosline (1988c) presented a free, damped oscillation of a tetiiered, but odierwise free-swimming animal. Their data are reproduced in Figure 2.9. There is no combination of parameters in die linear oscillator solution (Equation 2.4) which will exactly fit their data. The best fit (Table 2.2) to the second damped oscillation is shown in grey - a similar excellent fit can be made to die second damped oscillation, but it does not fit the first oscillation. In a linear system, the frequency and amplitude ratio of successive oscillations remains constant. However, as pointed out by DeMont & Gosline (1988c) in tiieir discussion of die data, both die frequency and the amplitude ratio in this case change with successive oscillations. They explained die observation as a change in die damping between high and low amplitudes: at large amplitudes, shear forces in die water flow around the bell would add to the internal energy losses in the mesoglea, while at low amplitude, die shear, and hence the damping, would be much reduced. This would result in less energy loss per cycle, and hence a higher amplitude ratio tiian predicted by the linear model. However, as can be seen from Equation 2.5, a decrease in die damping coefficient would cause an increase in die resonant frequency of the system, exactly the opposite of what is observed in Figure 2.9. One could argue tiiat the mass of fluid entrained by die movement of the bell was less in lower amplitude oscillations, but again, die prediction according to Equation 2.5 is that the frequency should increase. It does not, so a different explanation is required. Non-linear spring An alternate hypothesis is to be found in the non-linearity of the jellyfish spring. DeMont & Gosline (1988a) measured the meclianical properties of the mesogleal spring and showed tiiat its beliaviour is highly non-linear and strain-dependent. I will show in Chapter 5 tiiat tiiis strain-dependent stiffness creates a strain-dependent natural frequency - a bell that when hit harder, produces a higher tone. Using the model described in Chapter 5 and typical values for the force and spring functions, it is possible to predict die non-linear behaviour of the system, as shown by the blue line in Figure 2.9. The mechanism is as follows. At large amplitude, the stiffness of the system is high, and the recoil will happen at high frequency. At lower amplitude, the stiffness decreases, and die recoil frequency decreases. Likewise, die resistance to displacement by the spring is less at low amplitudes, so the ratio of peak amplitudes will increase. 3 Equation 2.5 is only true if the fluid load can be assumed not to contribute a non-linearity. (2.5) 32 5 -2 J , , , 1 1 . 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (s) Figure 2.9. Linear and non-linear harmonic oscillators. The data points are taken from DeMont (1986). Model parameters for both models are presented in Table 2.1. In the linear model (LHO - dashed grey line), the logarithmic decrement (see text) and frequency remain constant from one peak to the next. In the nonlinear model (NLHO - black line), the decrement decreases and the frequency increases from the first to the second peak. This phenomenon is described and explained in the text. The nonlinear model more accurately predicts the changes in amplitude and frequency observed by DeMont (1986). Parameter Linear Non-linear k (N/m) 1.85 x + i3006x3 b (Ns/m) 0.037 0.040 F(N) 0.0045 0.003 m (kg) 0.025 0.0145 w (rad/s) 2.0 2.5 Table 2.2. Fit parameters for Figure 2.9. The linear spring coefficient is taken from DeMont & Gosline (1988c). Force amplitude is measured in Chapter 6. Other parameters were obtained from the best fit of the model to the data in Figure 2.9. The spring function is arbitrary, but well within the range of observations made in Chapter 5. 33 The spring function and other parameters used to make die fit shown in Figure 2.9 are presented in Table 2.2. For comparison, DeMont & Gosline (1988c) reported a damping coefficient in the range 0.0618 -0.154 Ns/m, and a mass in the range of 0.012 - 0.031kg. The conclusion tiierefore is that die linear oscillator equation, although robust to many departures from its assumption of linearity, is not always sufficient to explain the behaviour of an oscillator. This is particularly so in systems which go tiirough large amplitude deformations. It is possible that many of the systems which have been considered non-resonant because the linear model could not be fit to them (e.g. squid Gosline & Sliadwick 1983a; dolphins: Pabst 1999), are in fact non-linearly resonant. Given the potential magnitude of energy savings to die animal if it does operate at resonance - DeMont & Gosline (1988c) estimated about 30% for die jellyfish, and I will show a that the savings could be much higher - it would seem that evolution ought to favour adaptations that make it possible. In the chapters that follow, I will consider the underlying mechanics of the non-linear elasticity of jellyfish, building a predictive numerical model of jellyfish mesoglea from measurements of the material properties of its parts. I will show that jellyfish of all sizes are able to obtain the significant energy savings of resonance (roughly 30% for large and mid-sized animals, but up to 80% for small animals) by swimming, as they do, in die resonant gait. 34 Chapter 3. The Modulus of Elasticity of Jellyfish Microfibrils. Introduction The mesogleal bell of Polyorchis is similar to tliat of all other medusae, in that it is a gel reinforced with a set of radially-oriented fibres (Gladfelter 1973). The energy required to refill the bell cavity is thought to be stored in the elastic elongation of the fibres (Bouillon & Vandermeerssche 1957, Gladfelter 1972, Weber & Schmid 1985, DeMont & Gosline 1988a). During deflation, the fibres are stretched by the change in shape of the mesogleal bell wall. The volume of the mesoglea does not change, so as the radius of the subumbrellar cavity gets smaller, the bell wall must get thicker, thereby stretching the fibres. The elastic recoil of the fibres draws the ex- and sub-umbrellar epithelia together, forcing the radius of the bell to increase. The fibres have recently been identified as microfibrils made of a protein homologous to mammalian fibrillin (Reber-Muller et al. 1995). Direct measurements of the mechanical properties of jellyfish microfibrils have not yet been made, but DeMont & Gosline (1988a) used Gladfelter's (1972) data on the density and cross-sectional area of Polyorchis fibres to predict on the basis of energy storage arguments that the modulus of the fibres should be about IMPa. This is well within the range of subsequent authors' measurements of the elastic modulus of inicofibrils in other organisms (lobster aortae: 1.06 MPa (McConnell et al. 1996), sea-cucumber dermis: 0.2 MPa (Thurmond & Trotter 1996)). In this chapter, I find the modulus of elasticity of jellyfish microfibrils to be approximately 0.40 MPa, less than DeMont & Gosline's prediction, but more than stiff enough to account for the energy necessary to refill die bell. The second component of the jellyfish bell is the gel, or mesogleal matrix. Several studies have been conducted on the chemical composition of the material (Chapman 1966, Bouillon & Coppois 1977, Weber & Schmid 1985, Reber-Muller et al. 1995), but there are few measurements of its mechanical properties. Alexander (1962) studied the creep behaviour of mesoglea, but over time frames of no relevance to the animal's swimming behaviour. DeMont & Gosline (1988a) studied the mechanical behaviour of mesoglea, both in isolated samples and in a novel intact animal preparation. They concluded that the overall tensile stiffness of mesoglea was between 400 and lOOOPa, but did not separate the contributions of the fibres and matrix. I present in this chapter the first compression tests of jellyfish mesoglea, and find tliat the compressive stiffness, and hence stiffness of the extracellular matrix, of the joint mesoglea to be approximately 130Pa. The bell mesoglea is stiffer in compression, approximately 352Pa, and the tensile stiffness along the fibre axis is approximately 1186Pa. 35 Materials & Methods Study animals Live P. penicillatus (Figure 3.1) were collected by SCUBA divers from the waters of Bamfield Inlet and Esquimalt Harbour, on die west coast of Vancouver Island, BC, Canada. Fibre diameter measurements were made at the Bamfield Marine Sciences Centre. Animals were kept there in running seawater aquaria until use. The remaining experiments were conducted at the University of British Columbia, so animals were shipped in chilled seawater, tiien held in a recirculating seawater aquarium until use. Aquaria in both locations were maintained at 11°C, and all animals were fed live Artemia every other day. Microscopy Fibre density To prepare samples for microscopy, the apex of the bell was first removed from the animal above the shoulder joint (Figure 3.1). The resulting ring of muscle and mesoglea was then sliced longitudinally along one side to lay the animal out flat. Next die flattened sample was laid on a microscope slide and cut into 3-5 mm strips with an unused razor blade. The sections were tiien laid on a freezing microtome, sliced edges down and up, such that the radial fibres were oriented parallel to the cutting surface, and shaven first on one side, tiien on the other, until they were 500uin thick. Finally the microtome was turned off and allowed to warm up. As soon as die frozen sample had thawed sufficiently to release itself from the microtome stage, it was transferred carefully to a new microscope slide and covered with a coverslip. Digital micrographs were taken of each sample using a video camera mounted on a Leitz Orthoplan interference contrast microscope, using 25x and 40x objectives and a first order red filter to enhance die contrast. Images were captured on a PC using a National Instruments 1024 video capture board and Lab View IMAQ software. Pixel dimensions were calibrated using micrographs taken at die same magnification of a set of bars of IOuin separation. Density was defined as the number of visible fibres intersecting a line across the micrograph, regardless of fibre diameter, divided by die widtii of the micrograph (40x: 640 px = 128 uin; 25x: 640px = 206um), multiplied by the original thickness of the sample (500 urn). Note that tiiis method is different from Gladfelter (1972), so his data were converted to enable comparison (see Table 3.1). Both calculations assume tiiat the shrinkage of the sample in preparation and handling was due only to the loss of water and matrix, and that few fibres, if any, were lost. Although it was impossible to confirm this assumption, it seems reasonable given the high degree of intertwining and consequent solid anchorage of the fibres in the exumbrellar epithelium and gastrodermal lamella. 36 Subumbrella Figure 3.1. Polyorchis morphology. The mesoglea is divided into two regions by die gastrodermal lamella. During contraction, since the muscle is only attached to the mesoglea at the per- and inter-radii, the bell first folds around die adradial joints, and only later in die contraction begins to stretch the radial fibres. AR, ad-radius; IR, inter-radius; PR, per-radius; P, peduncle; RC, radial canal; RF, radial fibre; G, gonads; JM, joint mesoglea; BM, bell mesoglea; M, manubrium; V, velum; RM, radial muscle; CM, circular muscle; GL, gastrodermal lamella. Photograph by the author, drawings modified from Gladfelter (1972). 37 Fibre diameters Samples were prepared in a fashion similar to tliat used in the previous section, but without the microtome. Approximately 3 mm duck sections of mesoglea were sliced off the strip of mesoglea using a new razor blade. The sections were laid sliced side down on a microscope slide and covered with a cover slip. They were allowed to dry down for 1 hour before measurements were made, in order to facilitate focusing of the microscope. Measurements of diameter were made using an optical micrometer mounted in one eyepiece of an interference contrast microscope, at a total magnification of 1875x. The micrometer was calibrated by measuring bars of known separation at the same magnification. Mechanical testing Two types of mechanical tests were done to characterise the elastic behaviour of jellyfish mesoglea in radial tension. The first were done in air with a slab of isolated mesoglea while the second were done underwater on an intact animal. Both experiments were done using an Instron testing machine and a custom-built load cell. Morphological measurements (bell height, shoulder height, margin diameter, and wall tluckness) were made using calipers before testing. Slabs of mesoglea were prepared and tested as follows. The apex was first removed by slicing the animal at the shoulder joint perpendicularly to its long axis. The resulting ring of muscle, skin and mesoglea was then sliced along one radial canal so that it could be laid flat on the moving stage of the testing machine (Figure 3.2 A). A 7mm x 7mm section of the stage (the lower grip) was covered with cyanoacrylate adhesive (Krazy Glue), and die mesogleal slab preparation laid over it such tliat one of the per-, ad- or inter-radii (Figure 3.1) was centred on the grip. The dimensions of the grip and sample were selected to ensure tliat the edges of the sample were sufficiently distant from the grip that they did not interfere with the experiment. The stage was then raised until the upper surface of the mesogleal slab contacted a second, identically sized grip, itself connected to die load cell at die top of the testing machine frame. The stage was raised a little further, until 20mN of force were applied to the mesogleal slab, to ensure tliat die glue set properly. The stage was lowered again after 10 seconds until the force returned to zero. This was set as die lower limit for the load cycles. The self-loading of the tissue made it impossible to make a meaningful direct measurement of the resting length, lo. Furthermore, die action of pressing the sample against the grip to set die glue forced the joint mesoglea out die wound at die shoulder, and also caused some irreversible compression of the bell mesoglea4. The value of 10 was dierefore back-calculated from die stiffness at high extension using a mediod identical to Lillie et al. (1994). A linear regression was fit to die loading curve at high extension, 4 20mN of force corresponds to approximately 400Pa of compressive stress on die sample. Compare dus to the 80-150Pa of tensile load typically applied during an experiment. 38 Instron stage Load cell B Figure 3.2. Mechanical testing apparatus. A. Isolated mesoglea preparation. A slab of mesoglea was laid on the moving stage of an Instron testing machine. Cyanoacrylate adhesive was applied to both plates. B. Intact animal preparation. Glue was applied to the bottom of the upper plate and top of the lower plate. 39 and the point at which the regression line crossed the x-axis (zero stress) was taken to be 10. Once 10 had been determined, the extension data were converted to engineering strain (e = Al /10). Engineering stress and strain are approximations to true stress (force / instantaneous area) and strain (extension / instantaneous extension). They are related to their true counterparts by: where unsubscripted symbols refer to instantaneous quantities, and subscripted ones to resting values. Engineering stress and strain are precise only for small extensions. The consequence of using engineering quantities is that die stiffness in compression will be overestimated, while tiiat in tension will be underestimated. However, die error introduced by using these small deformation approximations is less tiian die experimental uncertainty introduced due to the highly compliant, easily deformable nature of die materials under test. In order to measure die stiffness of the joint mesoglea, a second, intact animal preparation was designed (Figure 3.2B). A 7mm square plate of polystyrene was glued to an L-shaped rod suspended from the load cell. A second identical polystyrene plate was cantilevered from a post fastened through the base of plastic beaker to die Instron stage. The beaker was filled with 11°C seawater to a level just below die lower plate. Cyanoacrylate adhesive (Krazy Glue) was applied to die plate, and the jellyfish positioned over the plate as shown in the figure such tiiat one of the per-, ad- or inter-radii was centred on the plate. The glue was given twenty seconds to set before additional seawater was added to raise the level to just below the outer (now upper) edge of the mesogleal bell. Cyanoacrylate adhesive was applied to die bottom of the upper plate. The stage below the beaker was tiien raised until the upper plate came in contact with the exumbrellar surface of the jellyfish. The stage was raised a little further, until 20mN of force were applied to die jellyfish, to ensure tiiat the glue set properly. After ten seconds, seawater was added until the jellyfish was completely submerged. The stage was tiien lowered until die force returned to zero. As before, this was set as die lower limit for the load cycling. The resting thickness, 10, was calculated as above for die isolated In both cases, die sample was loaded in tension at lOmm/min to various strains between 5 and 40%, though if the latter was not enough to cause the sample to yield, experiments were continued at increasing strains until it did yield. Stress was defined as die load divided by the surface area of the polystyrene plates (engineering stress), and strain as the extension divided by the resting length, 10 (engineering strain). (3.1) cr eng slabs. Statistical tests were carried out following Zar (1984). Except where otherwise noted, results are given as mean +/- SE (standard error of the mean). 40 Results Fibre morphology Fibres were oriented more or less perpendicularly to the ex- and sub-umbrellar surfaces, and traversed most of the thickness of the bell, anchored in die exumbrella by intertwining witii the fibres there, as observed by other workers (Gladfelter 1972; Weber & Sclunid 1985; Reber-Muller et al. 1995). Near the gastrodermal lamella, they branch multiply into finer and finer fibres, presumably anchoring diemselves in the lamella, diough tius could not be discerned using the available microscope. Branching began about half-way across die thickness of the bell. Because of the fixation process (freezing & thawing), the sample tended to leak & shrink substantially, with a loss of radial thickness typically of about 40%. When viewed through the microscope, the fibres were slack (Figure 3.3). Because die fibres assume an irregular helical conformation when slack (Gladfelter 1972), it was not possible to measure their exact length from die micrograph. However, a rough estimate can be obtained by assuming the helical conformation to be circular. The length of a circular helix can be written: L = 2nR'N (3.2) where R' is the radius of a loop of the helix, and N is the number of loops. The axial length of the coil is given by: / = 4R/Vtan# (3.3) where R is die radius of the coil perpendicular to its long axis, and 0 is half the opening angle (see inset, Figure 3.3). The ratio of L to 1 is: L ITJR'N n I 4 AT? tan 0 2sin0 (3.4) From Figure 3.3, it is possible to estimate an average 9 of about 55° and R of about 15.5u.in, making die length ratio approximately equal to 1.96. The thickness of the outer mesoglea layer in Figure 3.3, and hence die average coiled axial lengtii of the fibres, is about 0.5mm, suggesting tiiat the total length, L, of the fibres was approximately 1.0mm. The total radial tiuckness of the collage shown in Figure 3.3 is 1.35mm. The animal's measured resting wall thickness was 3.0mm, indicating a 55% loss of radial tiiickness due to water loss. Thus if the water loss was uniform throughout the animal, the corrected tiuckness of the outer mesoglea was approximately 1.11mm, suggesting an in vivo pre-strain of the fibres of about 10%. This is obviously an educated guess, but it does partially confirm die speculation by previous authors that the fibres were likely pre-strained in vivo (Bouillon & Vandenneerssche 1957, Chapman 1958, Gladfelter 1972, DeMont 1986). 41 Figure 3 .3 . Collage of micrographs showing a complete cross-section of a jellyfish. Micrographs were taken using a video capture system and a 25x objective on an interference contrast microscope. Fibres can be traced from die inner edge of die exumbrellar epidermis to the gastrodermal lamella. Note die high degree of branching of die fibres at dieir ends, and die intertwining of the fibres witii die tissue of the exumbrellar epitiielium and die gastrodermal lamella. Also note the coiled, slack appearance of the fibres in the medial half of the bell mesoglea, and die absence of organisation in die fibres of the joint mesoglea. The black rimmed circular structures in the micrographs are air bubbles introduced during die tiiawing and transfer of die sample from the microtome to a microscope slide. Inset. Side view ofa coil. R' is the radius of one loop of the coil, R is die projected radius of die coil on the plane perpendicular to die long axis of the coil, 6 is the opening angle of die coil, 1 is die axial length of the coil, and L is die total length of the fibre. 42 Exumbrellar epidermis Bell mesoglea Gastrodermal lamella Joint mesoglea (Tear) Subumbrellar epidermis Slack, coiled fibres Location of sample Highly branched region 43 Fibre densities Figure 3.4 is a set of typical micrographs used to measure fibre densities. The distribution of fibre density over the cylindrical section of the bell is summarised in Table 3.1 (Jellyfish 1-6). Fibre densities were highest at the ad-radius, AR, followed by the per- and least at die inter-radii (PR and IR respectively). Consistent with Gladfelter (1972), the density of fibres was greatest near die gastrodermal lamella (subumbrellar side), and least near die exumbrellar side (Figure 3.3). The higher density near the gastrodermal lamella was due to die high degree of branching in that region. Only micrographs from the mid-thickness region of the bell wall, where fibres were mostly unbranched, were used in the calculation of the average fibre densities reported in Table 3.1. Gladfelter (1972) presented similar data, but used a different method for calculating fibre density. He did provide enough data however to make conversion possible: to obtain die data presented in die second half of Table 3.1 (Jellyfish 7-11), his density data were first converted to absolute fibre numbers, then divided by the cross-sectional area of his samples, namely 0.5mm x 1.0mm. The average fibre densities were 253, 240, and 410 mm"2 in die inter-, per-, and ad-radii, respectively. The overall average fibre density was 301 mm"2. Fibre diameters Typical micrographs used to measure fibre diameters are presented in Figure 3.5. 480 measurements were made of 120 fibres in 4 jellyfish - data are presented and summarised in Table 3.2. Sections of mesoglea were taken from inter-, per- and ad-radii, and no significant differences in fibre diameters ( A N O V A , p>0.05) were found between these regions. Furthermore, no significant differences were found between animals ( A N O V A , p>0.05), SO all of the measurements were pooled. The average unbranched fibre diameter was 3.03 +/- 0.33 ^in. This is substantially larger than the 1.5um reported by Gladfelter (1972) and the 1.8um reported by Weber & Sclunid (1985). Neitiier reported how the measurements were made, nor from animals of what size, nor did they give a range. Gladfelter's (1972) and Weber & Schmid's (1985) measurements were made on histological preparations, so it is most likely that a reduction in the water content of the fibres accounts for their smaller dimensions. 44 ' j • ' / • /• "•'' .'" 11 ! " ' / f • M Near exumbrella 44 fibres Mid bell 12 fibres Near gastrodermal lamella 64 fibres Figure 3.4. Micrographs used to measure the density of jellyfish microfibrils. Micrographs taken with a video capture system using a 40x objective on an interference contrast microscope. Samples were allowed to dry down for an hour before measurements were taken, to facilitate focussing the microscope. Fibres were branched at both ends, more so at the medial end, near the gastrodermal lamella A. near the exumbrellar surface. B. mid bell. C. near the gastrodermal lamella. Fibre density was determined by counting the number of fibres crossing the black line across the micrograph. Micrograph width: 640px = 128nm Depth of sample: 500|im. 45 Individual Bell height Fibre densities (mm1) (Jelly ID) (mm) IR PR AR 1 (B+x) 20 378 2(C+) 25 334 438 3(F) 20 268 4(J") 23.3 201 5(L) 20.7 249 201 402 6 (NT) 17.5 291 7 7 434 334 580 8 8 460 835 9 15 151 75 166 10 18 101 220 158 11 28 45 Average 253 240 410 Overall Avg 301 Table 3.1. Fibre densities. Data are die average fibre densities at die ad-, per- and inter-radii. Data for jellyfish 7-11 were presented by Gladfelter (1972) and converted as described in die text to the same format as die new data presented for jellyfish 1-6. Notes: * Individual was cut perpendicular to the long axis, in cross-section. + Micrographs taken using a 25x objective. * Thickness of slice was 300um IR: inter-radius; PR: per-radius; AR: ad-radius. 46 Figure 3.5. Micrographs used to measure fibre diameters, taken using a lOOx objective on an interference contrast microscope. Arrows show typical diameter measurements. Average diameter was 3.0 +/- 0.3 um. Individual Fibre diameter (xl0~*m) Mean Stdev n 21 2.88 1.03 AA 22 3.36 1.36 51 23 2.63 0.65 20 24 3.25 0.67 5 Overall 3.03 0.33 Table 3.2. Fibre diameters. Data are the average and standard deviations of fibre diameters measured using a lOOx objective in an interference contrast microscope. No significant differences were found between individuals, so the data were pooled to give an overall mean value. Overall standard deviation is relative to the animal means. 47 Mechanical properties of mesoglea Figure 3.6 shows a typical result of a radial test of an isolated slab of mesoglea. Negadve strains indicate that the sample is being tested in compression, while positive strains indicate tension. Because the joint mesoglea was forced out of the sample during die glue-setting step at the beginning of the experiment, the compressive stiffness measured was that of the dense fibre-reinforced bell mesoglea. In the plane transverse to the fibres, the tissue is isotropic, so the compressive stiffness of the mesoglea is also the tensile stiffness of the bell mesogleal matrix. Data for all jellyfish tested are summarised in Table 3.3. The average modulus of elasticity of the bell mesoglea, E m , was 352 +/- 39Pa (SE). The stiffness of die material was higher in tension, reflecting die fibre-reinforcement of the mesoglea in the radial direction. Data for the seven jellyfish tested in this manner are presented in Table 3.3. The mean stiffness of the mesoglea in radial tension, E L , was 1186 +/- 159Pa (SE). Typical results of die intact animal tests are shown in Figure 3.7. Data for die tiiree animals tested are shown in Table 3.3. The tensile stiffness of the intact preparation was similar to tliat of the isolated preparations, but, as expected, since the joint mesoglea is much less dense dian the bell mesoglea (Gladfelter 1972), the compression stiffness was much lower in die intact animal tests dian in die isolated preparation. The average stiffness of die joint mesoglea, Ej m , was 130 +/-1 lPa (SE). 48 Individual bell height wall thickness ID E M E L 0 20.7 3.1 2.19 258 826 P 19.2 2.7 1.83 451 1493 Q 19.3 2.3 2.09 488 978 R 17.1 2.6 1.99 441 1338 S 17.7 3 1.70 214 400 U 18.4 2.6 2.32 216 806 V 22.1 3.5 4.39 350 111 1241 X 16.5 2.4 2.89 149 1699 Y 17.6 2.9 4.08 400 131 1897 Average 352 130 1186 SE 39 11 159 Table 3.3. Elastic moduli of jellyfish mesoglea in radial tension. Dimensions in mm, stiffnesses, E i ; in Pa. E m is the compression stiffness of the bell mesoglea during isolated tissue tests, Ej m is die compression stiffness of the joint mesoglea during intact animal tests. E L is die stiffness of the system in tension, and represents die contributions of both the matrix and fibres. There was no discernible difference in E L between the two methods. To obtain lo, I fit a straight line through the highest reasonable loading slope on a graph of stress as a function of extension, and tracked it to zero stress. The abscissa was taken as 10. This usually lay somewhere near the inflection point in die loading curve. SE: Standard error of the mean. 49 Radial Tensile Test - Isolated Mesoglea Preparation Figure 3 .6 . Typical stress-strain behaviour ofa slab of isolated mesoglea in the radial direction. Negative strains represent compressive loading, while positive strains indicate tension beyond die resting thickness. The sample was compressed below its resting tiiickness during mounting, with the result tiiat the fibres, normally pre-strained in vivo, were slack. The joint mesoglea was removed in this preparation, so it was possible to measure the stiffness, E m , (dashed line), of the bell mesoglea alone. The solid line shows the parallel stiffness, E L , which includes contributions from matrix and fibres. Data for 7 jellyfish tested in this manner are summarised in Table 3 .3 . Zero strain was determined by regressing from the large strain data as discussed in die methods section of the text. 5 0 Radial Tensile Test - Intact Animal Preparation ro co 0 00 Figure 3.7. Typical stress-strain behaviour of intact mesoglea. Negative strains represent compressive loading, while positive strains indicate tension beyond the native thickness. Because there was no loss of joint mesogleal tissue in this preparation, the slope of the dashed line is die stiffness of the joint mesoglea, Ejm , while die slope of the solid line is again the radial tensile stiffness parallel to die fibres, and includes contributions from die matrix and fibres. Data for 3 jellyfish tested in this manner are summarised in Table 3.3. As in the previous figure, zero strain was determined from a regression through the large strain data. 51 Discussion Since there is no muscle to open the bell, it is obvious tliat the jellyfish mesoglea must be able to store enough energy to power the refilling of the subumbrellar cavity. This energy must be stored in elastic deformation of the tissue. Previous authors have noted die presence of radially oriented fibres in die mesoglea and attributed to them its ability to store elastic energy (Chapman 1953, 1959, Bouillon & Vandenneerssche 1957, Mackie & Mackie 1967, Gladfelter 1972, Weber & Schmid 1985, DeMont & Gosline 1988a). Gladfelter (1972) first proposed a mechanism by which they might act, noting that the diickness of the mesogleal bell increased as die animal contracted its muscle, thereby stretching die radial fibres. The stiffness of the radial fibres can be calculated from die tensile stiffness of the bell mesoglea by accounting separately for contributions by the fibres and matrix to die overall elasticity (McConnell et al. 1997): a = EfVf£f+Em(\-Vf)sm (3.5) where the E, are the moduli, the e, are the strains of the fibres (f) and matrix (m), and V f is the unitiess volume fraction of fibres in the mesoglea. Since individual fibres traverse die entire mesogleal wall (Figure 3.3), the strains on the fibres and matrix are identical. Rearranging Equation 3.5 and substituting in die combined stiffness, E L , of the mesoglea: E, -Em(\-Vf) E = __L m\ H Assuming an average fibre density of 301mm"2, an average fibre diameter of 3.03urn, an average E L of 1186Pa, and an average E m of 352Pa, tiien the average fibre modulus is approximately 0.40 MPa (range 0.20 to 0.74 MPa)5. For comparison, DeMont & Gosline (1988a) estimated IMPa, and McConnell et al. (1996) measured die stiffness of microfibrils in lobster arteries to be 1.06MPa. Note tliat both of tiiese values were based on fibre diameters measured from histological prepared samples, that is, samples from which all of the water had been removed. The modulus reported here is substantially lower since it is based on the diameters of wet fibres. If I had used Gladfelter's (1972) microscope measurement of 1.5u.m or Weber & Schmid's (1985) SEM measurement of 1.8um, I would have obtained a modulus of 1.09MPa. 5 This error estimate was derived using the mean +/- one SE for die stiffnesses, an estimated 10% measurement error for the fibre diameter, and a 20% error in the fibre density. 52 In his analysis of the deformation of the animal, Gladfelter (1972) showed that the presence of the joint mesoglea substantially reduced the radial strain relative to a hypothetical unjointed animal of otherwise identical resting dimensions. He concluded tiiat die joints existed to allow the animal to make a contraction of given magnitude for less input force. DeMont & Gosline (1988a) measured die non-linear stiffness of the intact mesogleal bell. They took Gladfelter's argument further, suggesting that the non-linear elasticity allowed the muscle to power die refilling stage by storing energy at a time in die jet phase when it is not useful for generating dirust (DeMont & Gosline 1988b). With the measured stiffness of die fibres, it is possible to determine whetiier die fibres alone are sufficient to account for the energy required to refill the bell. Elastic energy (W) is die integral of the elastic restoring force (F) over the distance stretched (x): W = JFdx (3.7) Expressed in terms of stress (a) and strain (e), this becomes: W = ja(s)Al0de (3.8) where A is die area over which die force is applied and 10 is die resting wall thickness. The area of interest here is die inner surface of the bell, A u , where: Au = 2nrmhs (3.9) and rm is die radius of the subumbrellar cavity and hs die shoulder height of die jellyfish. For die jellyfish in Figure 3.7, die total subumbrellar area is 3.3 x 10"4 m2. Gladfelter (1972) described diree regions in the bell: ad-, per-, and inter-radii. Assigning equal areas (Aj) and initial tiucknesses (l0,i = x) to die diree regions, the integral becomes: regions.! Q where die ej's are die local radial strains in each region. For the animal in Figure 3.7, each subarea (A;) is 1.1 x 10"4 m2 and the initial tluckness is 2.9 x 10"3 m. The upper limits of integration are obtained from Gladfelter's (1972) measurements of die radial strain in the three regions. These were: 36%, 34% and 16% in die per-, inter- and ad-radii, respectively (see Figure 1.3). The functional form of cr(e) is given by 53 Equation 3.5. Neglecting the contribution of the mesogleal matrix for the moment, substituting into Equation 3.10 and simplifying, gives: W=l-AurEfVf 0.34 0.36 0.16 = 3.63xl0" 5 y (3.11) The resilience of Uie mesoglea bell was calculated by DeMont & Gosline (1988a) to be 0.58, so die total energy tiiat can be released from die fibres is 2.10 x 10"5J, which is sufficient to meet die energy estimated by DeMont & Gosline (1988b) to be required to refill the bell. The model assumes that all of the fibres are aligned perfecUy with the local stress axis, and that they are all strained maximally. This will of course not be the case generally, so the total energy available will be somewhat less. The error associated with these assumptions would liave to be greater than about 24% in order for die energy storage in the fibres no longer to be sufficient to meet DeMont & Gosline's (1988b) minimum estimate of 1.7xl0"5J to refill die bell. I conclude tiierefore that die fibres can and do store the energy required to refill die bell, and tiiey are stiff enough to do so during the second half of the jet stroke, after die bell is maximally folded, when die tiirust generation efficiency of the muscle is diminished. In die next chapter, I will consider die geometry of bell deformation in more detail, and look into die contributions of die joint and bell mesogleal matrices. 54 Chapter 4. From Mesoglea to Jellyfish Introduction From an engineering perspective, a jellyfish is essentially a thick-walled cylinder, capped at one end, and made of a fibre-reinforced composite material. Fibre-reinforced composites are common in biology. They represent die most efficient use of material to create structures that are both flexible and stiff: very small volumes of reinforcing fibres are required to create significant stiffness (e.g. Gosline & Shadwick 1983a). In the soft-tissue literature, arteries are probably the best-studied variation on a thick-walled cylinder (e.g. Fung 1993, Shadwick & Gosline 1985). Elastic protein molecules are arranged circumferentially in the artery wall to reinforce the artery against the internal pressures created by the heart. Jellyfish are however very different from arteries in one important regard. While arteries are designed to witiistand hyperinflation, jellyfish are designed to resist deflation. This difference in die roles of the two structures is evident in die orientation of their reinforcing fibres. In arteries (and indeed in most cylindrical engineering structures), the fibres are arranged circumferentially, while in jellyfish tiiey are radially oriented (Gladfelter 1972, 1973, Weber & Sclunid 1985). This is die jellyfish's solution to die problem of compressive loading. Widiout solid structures such as bone, there is no efficient biological means of storing energy in compression. In tension, however, long-chain molecules can be made into particularly good tensile springs. The decreasing volume of the subumbrellar cavity loads die mesogleal bell in circumferential compression, but tiirough die Poisson effect, die system is also loaded in tension in die radial direction. This stretches the fibres and stores die energy required by the jellyfish to refill its subumbrellar cavity. In their study of the elastic behaviour of jellyfish mesoglea, DeMont & Gosline (1988a) found the material to be highly non-linear, a characteristic reminiscent of arteries. The non-linearity of arteries has been proposed to be a design to allow smooth flow of blood at low pressure but resist bursting at high pressure. In the jellyfish, die non-linearity serves a similar but opposite role. The low stiffness at small strain allows almost all of die force generated by the swimming muscle to be converted into useful forward thrust, while the high stiffness at large strain makes it possible for die animal to store enough energy during die final stage of the jet phase to power the refilling of the bell cavity. The non-linear stress-strain behaviour of vertebrate arteries is the result of elastin and collagen molecules acting in parallel - die compliant elastin provides some resistance to inflation at low pressure, but as die pressure increases, die highly kinked collagen fibres are straightened out, then stretched, stiffening die structure significantly at high pressure (Roach & Burton 1957, Wolinsky & Glagov 1964). In invertebrate arteries, because there is no elastin, a different mechanism is required. Shadwick & Gosline (1985) describe 55 an elastomer with non-linear properties in the octopus aorta, while in the lobster aorta, reorientation of microfibrils has been proposed to provide the non-linearity (McConnell et al. 1997). The idea is that as the pressure increases and the artery is stretched in die circumferential direction, the initially randomly oriented microfibrils become more parallel to the direction of stress, and hence increase the stiffness of the material at high strains. Similarly to the lobster aorta, jellyfish mesoglea is reinforced with microfibrils (Reber-Muller et al. 1995) which are linearly elastic (McConnell et al. 1996). The non-linear elastic beliaviour of the jellyfish mesogleal bell in deflation is die result of its complex geometry. The mesoglea is a mucopolysaccharide gel contained within an external epithelium. A layer of cells called the gastrodermal lamella divides the mesoglea into two regions. The outer layer is reinforced with radially-oriented fibres while the inner layer is almost or completely devoid of such fibres (Chapman 1953, 1959, Bouillon & Vandenneerssche 1957, Mackie & Mackie 1967, Gladfelter 1972, Bouillon & Coppois 1977, Weber & Schmid 1985). In cross-section, the gastrodermal lamella is roughly octagonal in shape, while die sub- and exumbrellar epithelia are circular (Figure 4.1). This creates a set of wedge-shaped regions of inner mesoglea, referred to as joints by previous authors (Bauer 1927, Gladfelter 1972, DeMont & Gosline 1988a). During contraction, the muscle first causes die bell to fold around these joints, tiien it compresses the mesoglea circumferentially. During tiiis compression the mesoglea expands radially, stretching the mesogleal fibres (Gladfelter 1972). DeMont & Gosline (1988a) devised a technique for measuring the elastic behaviour of the intact locomotor structure of the jellyfish. They glued a jellyfish by its margin to a sheet of Plexiglas and pumped water in and out of the subumbrellar cavity, while simultaneously measuring the pressure in the cavity. The passive elasticity of die mesogleal bell resisted the change in volume in much the same way as it does in free swimming. I use a similar device, built with updated components, to study the elastic behaviour of the mesogleal bell. The material properties of mesoglea (matrix and fibres) tiiat were measured in die previous chapter will now be incorporated into a numerical model of die jellyfish to predict the pressure-volume behaviour of the mounted jellyfish preparation. The excellent fit of the model to experimental data will confirm that the measurements of the fibre and matrix stiffness in the previous chapter were correct. Model Transversely isotropic composite material Because the fibres are essentially parallel, the mesoglea can be modelled as a transversely isotropic material (Spearing 2001). That is, die material is considered to be isotropic in the two directions perpendicular to the fibres, but the modulus in those directions will be very different from the modulus in die direction parallel to the fibres (Figure 4.2). The material is considered to be homogeneous, since the fibre densities throughout the bell were found in die previous chapter not to differ signficantly. 56 P Figure 4.1. Jellyfish geometry - thick walled cylinder with radial fibres (RF) & longitudinal joints (JM). P, peduncle; BM, bell mesoglea; EU, exumbrellar epithelium; SU, subumbrellar epithelium; AR; adradius; IR, interradius; PR, perradius; CM, circumferential (swimming) muscle; GL, gastrodermal lamella. Drawing adapted from Gladfelter (1972). 57 A. B. Figure 4.2. The jellyfish as a transversely isotropic thick-walled cylinder. A. Transversely isotropic material. Reinforcing fibres are oriented radially in the bell wall of the jellyfish. The mesoglea is modelled as an isotropic material in the other two directions. B. Thick-walled cylinder model, showing the radii and stress components. 58 To model the elastic behaviour of the material, we apply a generalised form of Hooke's law, namely: where e,j and aki represent die strain and stress components, respectively, in tensor form, and Syu is the compliance tensor, which collects die moduli together in matrix form. Spearing (2001) derives the components of the strain tensor from geometrical arguments and the Poisson effect. A strain in the fibre cr direction will create strains in the transverse directions, so for an applied strain e — : E L = ea = -yae„ (4.2) where die coefficient v L T is the Poisson ratio. Likewise, applied strains in die transverse plane (JM CT (£gg = — ,£zz = ——) will create strains in die fibre direction: Srr = -Wee = -VTL£* (4-3) Similarly, within the transverse plane, applied strains in one direction will create strains in die other: e„ = -v„ea and ea = -v^s^ (4.4) There will also be shear stresses, given by: *«=2GLTsn (4.5) °7* ~ ^-GjjSfo The isotropy in die transverse plane requires that: G n = ( E j , (4.6) 2(1 + ^ ) and die first law of diennodynamics requires diat: 59 vLTET - vTLEL (4.7) Gathering all of the above terms together, the tensor equation becomes, in matrix form: £rr £<k £r9 1 -VLT EL EL EL ~ VTL 1 — VJJ Eij. -VTL i ET ET Ely 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2G LT 0 0 0 0 0 1 2G LT <*90 (4.8) The stress components, 0 j j , for a thick-walled cylinder (Figure 4.2), capped at both ends6, under uniform internal pressure P, are tabled in Young (1989): rj =-pr7X-rt) r2[r2-r2) uee 2 2 2 r \r —r 1 \ e m / Pr2 °* = ( 2  m 2\ (4.9) (4.10) (4.11) Tlie hoop strain is the component of interest here. Substituting die above in die tensor equation, reducing and keeping only the circumferential component, gives: 2ntsr Pr' 2nr r.2(l + yn.) + r T 2 ( l - v i T ~VTL) 2 ( 2 T\ (4.12) 6 One end is die apex of the bell, and die other is created by gluing die margin of the bell to a sheet of Plexiglas. These conditions will be adjusted in die next chapter to reflect the geometry of the free-swimming animal. 60 The value of v-n, is unknown, but can be calculated using Equation 4.7: Gladfelter's (1972) data on the deformation of the bell in deflation provide an effective v L T = ^ / S e e = 0.44/1.403 = 0.31 (Figure 1.3B), and E L was measured in Chapter 3. In die transverse plane, mesoglea is a homogeneous water-based gel. The Poisson ratio in tiiat plane, vn, will therefore be approximately equal to 0.5 (Wainwright et al. 1976) Furthermore, the volume of the mesogleal bell remains constant on a time scale relevant to swimming contractions, and in both free-swimming (Gladfelter 1972) and in the mounted case (DeMont & Gosline 1988a), the length of the bell does not change. Therefore the outer radius, r^ can be calculated from the inner radius, rm, from: (4.13) Jellyfish geometry A cross-section of Polyorchis is shown in Figure 4.1. The bell wall is made of two different layers. The inner layer is die joint mesoglea contained between the subumbrellar myoepithelium and the gastrodermal lamella. The outer layer is the bell mesoglea, extending from the gastrodermal lamella to the exumbrellar epidielium. The inner layer is made of an isotropic, almost fluid gel (Gladfelter 1972), while die outer layer consists of a higher density gel, isotropic in the circumferential and longitudinal directions, but reinforced with fibres in the radial direction (Gladfelter 1972, Weber & Schmid 1985). Deflation The complex deformation geometry of die bell in deflation is modelled here in two stages. As the volume of the subumbrellar bell cavity is reduced from the resting state to some critical value, the element resisting the negative cavity pressure7 is the compressive stiffness of the joint mesoglea as the bell "folds" around die adradial joints. For this stage, the animal is modelled as an isotropic thick-walled cylinder with low stiffness, E c , representing the resistance to circumferential compression in the adradial joints. At higher deflation strains, beyond a critical strain, ec, the stiffer, fibre-reinforced bell mesoglea is recruited once the "folding" is complete. Further circumferential compression translates into radial tension, as the overall resistance to increasing negative pressure is provided by the combined action of die joint and bell mesoglea. Because of the presence of radially oriented fibres, an anisotropic model is required for tiiis stage. The animal is modelled in tiiis stage as a transversely isotropic thick-walled cylinder, with transverse stiffness equal to die sum of die stiffnesses of the bell and joint mesogleal matrices (ET = E m + E c ) , since 7 For die purposes of the model, die pressure is defined as negative if the radial force on die mesogleal surface (as opposed to die subumbrellar surface) of die circumferential swimming muscle is directed into die cavity (ie. towards the long axis of the animal). This definition ensures tiiat both die mounted preparation, driven by an external actuator creating a negative pressure in the bell cavity, and the free-swimming animal, driven by muscles creating a positive pressure in the cavity, but a negative one on the inner surface of the bell wall, are treated in the same fashion. 61 they are acting in parallel to resist circumferential compression, and fibre stiffness, Ef, which combines with the bell mesogleal matrix stiffness to provide an overall stiffness in the radial direction of E L . Inflation In inflation, the jellyfish behaves like an artery, with non-linear elasticity due to the sequential recruitment of circumferentially oriented structures such as the ex- and sub-umbrellar epitiielia, the kinked gastrodermal lamella, and the mesogleal matrix itself. A model such as that developed by Shadwick & Gosline (1985) will likely produce a satisfactory fit to the data. In Chapter 2,1 showed that in the resonant gait, the animals are hyperinflating by about 3 to 5% (Figure 2.4). Nonlinearly elastic transversely isotropic thick-walled cylinder model The final mathematical description of the beliaviour of the combined system can be derived from Equation 4.12 and rearranged to predict the cavity pressure (P) as a function of the radii (r^ . P = F F — ^gg^c 2 rl F E — r re2(l + v) + rt2(l-2v)\ (re ~rm ) rt (l + v r L ) + r r (1-VTX - V T T ) . (4.14) Incorporation of Equation 4.13 into Equation 4.14 yields the predicted quasi-static pressure-volume relationship for die intact bell. Note tiiat the thick-walled cylinder model is an engineering approximation to die behaviour of real objects and is based on the assumptions tiiat strains are small, the cylinder does not crumple, and the material is homogeneous and isotropic. The anisotropic, crumpling, large strain model presented in Equation 4.14 is limited by its violation of these underlying assumptions. More precise modeling of die structure would require die use of finite element methods. Nevertheless, for the purposes of developing a hypothesis, die analytical model presented in Equation 4.14 will suffice. Materials & Methods Study animals Live Polyorchis penicillatus were obtained from the waters of Esquimalt Harbour and Bamfield Inlet, on the west coast of Vancouver Island, British Columbia, and kept in running sea-water aquaria until use. They were fed \i\cArtemia daily. Only healthy, vigorously swimming animals were selected for study. Measurements were made of each animal's bell height, shoulder height, velar diameter, and wall thickness, as described in Chapter 2, before further experiments were carried out. 6 2 Apparatus Figure 4.3 shows the experimental apparatus used in tius and die next chapter to measure the behaviour of die intact bell. The method and apparatus are similar to DeMont & Gosline (1988a), but built using updated equipment. After removal of their tentacles and vela, animals were glued using cyanoacrylate adhesive to a circular nylon disk mounted on a Plexiglas sheet. The disk served to maintain the base of die bell in a circular shape while the glue set, creating a leak-proof seal. This prevented the margin from contracting as it does in the free-swimming animal. The matiiematical model developed in the previous section takes this into account dirough die boundary condition Uiat both ends of the cylinder are capped. The Plexiglas plate with its mounted jellyfish was placed in an aquarium, jellyfish side down, and supported 3cm off the bottom. Air bubbles inside the jellyfish were tapped out dirough a hole bored tiirough die Plexiglas sheet and dirough the centre of die nylon disk. Once the air bubbles had been removed, a support stem was inserted and its length adjusted so that the animal was held at its resting shape. A collar soldered to die stem about 2.5cm from its sealed end prevented its insertion too far into the bell cavity. The joint between die support stem and the Plexiglas was sealed with an o-ring set into die nylon disk, and vacuum grease was applied between the collar and die Plexiglas. The support stem consisted of a capped 6.4mm brass pipe tiiat had a set of 2 mm holes drilled dirough its side wall. A stainless steel catheter ran dirough the centre of die stein, silver soldered to die stem near the cap and where it exited die stem upstream of the collar. A stainless steel machine screw was threaded dirough a hole in die cap, so tiiat the overall length of the support stem could be matched to die resting lengtii of die jellyfish. The upstream end of the support stem was attached to a set of plastic hoses, in turn connected to die actuator. A computer-controlled solenoid relief valve was provided on a branch off the tubing to allow die system to return to ambient pressure after each trial. A Ling Power Systems actuator was used to drive a brass piston (1.3cm I.D.), which forced the water tiirough the tubing, tiien in and out of the mounted jellyfish. The system was feedback controlled by computer using a PID algoridun (National Instruments, Austin, TX). Pressure inside die jellyfish bell was measured using a variable inductance pressure transducer (Model DP-103, Validyne Engineering), connected by plastic tubing to die stainless steel catheter in the support stem. The transducer measured die change in inductance due to die deflection of a stainless steel membrane (Validyne #8-24) sandwiched between two coils. Its sensitivity was rated at 2.54cm H 2 0 full scale. The signal was routed through an AC-DC converter (Model CD-101, Validyne Engineering, Northridge, CA), 63 Pressure Transducer Figure 4.3. Experimental apparatus. The actuator drove a piston which forced water through plumbing into and out of the bell cavity of a jellyfish which had been glued by its margin to a sheet of Plexiglas. A support stem inserted into the cavity ensured that the jellyfish did not crumple longitudinally. A differential pressure transducer measured the pressure inside the bell cavity through a catheter inserted through the support stem. The reference side of the pressure transducer was open to the surrounding fluid. A solenoid valve on a T off the actuation plumbing allowed the system pressure to be equilibrated between trials. A camera and video dimension analyser (VDA) recorded the diameter of the subumbrellar cavity. The actuator was controlled by a PJD algorithm. Data was collected on the pressure, piston displacement, actuator input signal, and subumbrellar cavity diameter, at 100Hz. 64 which had a 3Hz electronic fdter built into it, and dience to tiie controlling computer. The volume of water injected or removed from the jellyfish was measured using a strain gauge built into die actuator. The strain gauge signal was amplified and input to die controlling computer. This signal was also the controlling variable for the PED algorithm which ensured that the actuator arm oscillated sinusoidally. The strain gauge output was calibrated to the movement of the piston, then muldplied by the internal cross-sectional area of the piston to convert it to a volume signal. Changes in die diameter of the subumbrellar cavity were monitored using a video camera and a video dimension analyser (VDA: Model 303, Instruments for Physiology and Medicine, San Diego CA), which output a voltage proportional to die separation of two contrast edges on the video image. This voltage was calibrated using an image of a ruler positioned next to die jellyfish. Software written using the Lab View data acquisition package (National Instruments, Austin TX) both controlled the actuator and equalizing solenoid, and sampled die pressure, actuator displacement, actuator input voltage, and bell diameter from die VDA, at 100 Hz. Calibration The displacement transducer (strain gauge) was calibrated using a graduated pipette attached by plastic tubing to the end of die piston. Dish soap was added to the top of the water column in the pipette to minimise the size of the meniscus, which was tiien followed using a video camera and the VDA. Suitable contrast was obtained by adding red food colouring to die top of the water column. The pressure transducer was calibrated similarly, using a manometer and metre rule with the video camera and VDA. Results Pressure-volume data Quasi-static measurements of pressure and volume are presented in Figure 4.4. Experiments were all conducted at 0.1Hz. Nine jellyfish were tested, ranging in size from 15 to 30 mm (bell height). The resting volume was defined as the volume at die inflection point of the 3rd order polynomial fit to the data. Zero pressure was also defined at die inflection point. The colours in the figure correspond to individual jellyfish; dashed and solid lines refer to different amplitude trials. The thick solid line is the global average, calculated as follows: A tiiird order polynomial was fit to die data and the curves lined up at their inflection points. A new polynomial was then fit to each dataset. Once all of the data had been fit, each polynomial was evaluated over its volume range. The global average was obtained by fitting a tiiird polynomial function to the point by point average of all curves. Polynomial fit coefficients are presented in Table 4.1. 65 Pressure - Volume, Mounted Jellyfish Preparation Figure 4.4. Quasi-static pressure-volume behaviour of mounted jellyfish preparation. Colours correspond to individual jellyfish; dashes or solid lines refer to different experiments. Thick red line (indicated by arrow) is the global average over all jellyfish tested. Fit parameters are presented in Table 4.1. Typical data used to generate these fits are presented in the lowest frequency panels of Figure 5.7. 66 Jelly / Expt. ao ai a 2 a 3 a 8 0 6.60E+06 6.11E+12 6.46E+19 c35 0 7.86E+06 -1.12E+13 6.70E+19 d47 0 1.02E+07 -2.62E+13 2.08E+20 d66 0 1.09E+07 -2.29E+13 1.68E+20 e27 0 2.28E+07 -2.26E+13 2.51E+20 e57 0 1.11E+07 -2.07E+13 1.98E+20 f 51 0 6.54E+06 -9.41E+12 7.15E+19 f 68 0 8.55E+06 -8.44E+12 6.31E+19 g32 0 8.46E+06 -3.69E+13 2.24E+20 h 16 0 1.64E+07 -9.64E+10 3.59E+19 h 17 0 1.72E+07 5.90E+10 4.08E+19 i 15 0 1.69E+07 6.20E+12 7.33E+19 k 9 0 4.92E+07 3.09E+12 3.79E+20 k57 0 6.07E+07 -1.92E+13 7.33E+20 Average 0 1.81E+07 -1.12E+13 1.84E+20 Table 4.1. Tiiird order polynomial fits to the pressure-volume data in Figure 4.4. Functional form: P = ao + a, V + a 2V 2 + a 3V 3. The value of die pressure at the inflection point in die curve was set as zero pressure -hence ao = 0 for all trials. Data labelled Average are die coefficients of the global average curve in Figure 4.4. 67 The non-linear behaviour of the structure is evident in die figure. As the bell was deflated, die pressure change was small at first, reflecting die folding of die bell around the adradial joints. As the volume was reduced further, die pressure in die cavity began to drop significandy as the bell mesoglea and its fibres were recruited. In inflation, die pressure rose immediately as die animal's muscle and epidielia were stretched circumferentially. At high inflation, die animal also tended to extend longitudinally - when this occurred, the driving amplitude was adjusted until the peduncle remained in continuous contact with the support stem. Fitting the model The model and data are in excellent agreement, as shown in Figure 4.5. Similar quality fits were obtained to data from all experiments. The joint, transverse and radial stiffnesses (E c, E T and EL) required to make the fit are well within die ranges of the experimental measurements in Chapter 3 of Ejm , (Ejm + Em), and E L , respectively (Table 4.2). As observed by Gladfelter (1972), die joint mesoglea is much more compliant than the bell mesoglea. Circumferential joint stiffness was on average about 3x less than the transverse (or matrix) stiffness of die bell mesoglea. The critical strain, s c , was on average about 10.6% deflation (rc = 0.894 r0). This compares favourably with Gladfelter's (1972) observation that the inner circumference of the bell decreases by approximately 14%, when die muscle is fully contracted, die joints are fully compressed, and die fibres fully stretched. Note that die model at this point does not incorporate any means of dealing widi die non-linearities in inflation, which cause the animal to be stiffer in inflation dian die isotropic diick-walled cylinder model would predict. This is very similar behaviour to an artery and could be modelled as such. A non-linear model of E T could be incorporated in die model and fit to die data to describe empirically the behaviour of jellyfish mesoglea in hyperinflation. Discussion The mesogleal bell of the jellyfish is an important part of its locomotor system. As die only available antagonist to die action of the swimming muscle, it must be able to store enough elastic energy in deformation to power the refilling of the body cavity after die muscle relaxes at die end of the jet phase. Its watery consistency, required for buoyancy, nutrition and flexibility (Chapman 1966), is however not conducive to storing energy in compression. The jellyfish's solution is to store die energy in tension tiirough die Poisson effect - hence die reinforcing fibres aligned radially in the bell wall which greatly increase die stiffness of the material, allowing it to store quite substantial amounts of energy. The stiffness of the mesoglea must not however interfere witii die efficient creation of forward tiirust. The non-linearity inherent in die deflation of a cylinder, even an isotropic one made of a linearly elastic 68 30 20 10 ro CL (D ^ -10 » CD ^ -20 -30 -40 H -50 -0.4 Thick-Walled Cylinder Models Data (Jellyfish K) Isotropic cylinder Anisotropic model -0.3 -0.2 -0.1 o.o 0.1 0.2 0.3 Volume (x 10~6 m3) Figure 4.5. Experimental data and fit of transversely isotropic thick-walled cylinder model Qiquation 4.14). Data points are from experiment k 57. Solid line is isotropic thick-walled cylinder model with stiffness E c = 300 Pa. Dashed line is anisotropic thick-walled cylinder model with radial stiffness E L = 1186 Pa and circumferential stiffness E T = 600 Pa. Note die two lines exactly overlay one another until the critical volume. The critical volume is about 83% of resting volume, corresponding to a circumferential strain of about 11%. The model fits die data very well in deflation. No attempt was made to model die inflation side - hence the lesser quality fit there. 69 Jelly / Expt. E c E T E L v c n t a 8 200 500 1186 0.83 c35 180 500 1186 0.81 d47 300 800 1186 0.78 d66 270 700 1186 0.78 e27 280 ' 550 1186 0.84 e57 200 550 1186 0.80 f 51 80 180 1186 0.70 f68 80 160 1186 0.70 g32 220 700 1186 0.80 h 16 480 750 1186 0.90 h 17 550 1000 1186 0.86 k 9 250 500 1186 0.80 k57 300 600 1186 0.83 Table 4.2. Stiffnesses (in Pa) and critical volume (proportion of resting bell cavity volume) required to fit the anisotropic model to the data. Note, the transverse stiffness in the anisotropic model is the sum of the compressive stiffness of the joint mesoglea, E j m , and the stiffness of the matrix, E m - i.e. E T = E„, + E c . The model was not very sensitive to changes in E L , as long as it exceeded about lkPa, so the average E L reported in Chapter 3 was used for all animals. 70 material, is useful in this regard since it presents the muscle with a low stiffness during the early part of the contraction, stiffening only as the volume gets smaller, and the efficiency of thrust development drops off. Indeed, many jellyfish species lack die sophisticated joint system of Polyorchis (Gladfelter, 1973). The animals that do have the joint system tend to be more active swimmers (Gladfelter 1973). The advantage conferred to the animals by the joints is an increased non-linearity. In jointed animals, the initial stiffness opposing die muscle during contraction is very much lower, permitting an increased forward thrust performance. This does however require a higher stiffness at the end of the jet phase in order to store enough energy to refill the bell. This is accomplished in part through a higher density (and hence stiffness) bell mesoglea, and also through a higher degree of fibre development in the more actively swimming species (Gladfelter 1973). The model presented in this chapter successfully predicts die quasi-static elastic behaviour of the mesoglea in deflation. It does so based on experimental measurements of the ineclianical properties of the component parts of the structure, and on a simple model of the animals' complex geometry. In the next chapter, I will expand die model to predict the dynamic behaviour of die mounted jellyfish preparation, tiien compare the prediction to observations of the real system. I will tiien relax die boundary conditions of the model to predict the elastic behaviour of the free-swimming jellyfish. 71 Chapter 5. Dynamic Elastic Behaviour (Resonance) Introduction Mechanically, the jellyfish is a system composed of a spring and a mass - in other words, an oscillator. Several workers (Bouillon & Vandermeerssche 1957, Chapman 1958, Gladfelter 1972, Weber & Schmid 1985, DeMont & Gosline 1988a) have described die radially-oriented fibres in the mesoglea and concluded diat they must be the springs opposing die muscles. However, Gladfelter (1972) noted that the frequency of oscillation was not constant, as would be expected of a simple harmonic oscillator with constant stiffness. He concluded tiiat the muscles must be modulating the effective stiffness of the bell structure. I have shown elsewhere (Chapter 2), based on research by Spencer & Satterlie (1981), that the muscles are not likely active during die recoil stage, so Gladfelter's observation of variable recoil frequency must have some other explanation. DeMont & Gosline (1988a) devised a method to measure die pressure-volume behaviour of the intact locomotor structure and showed it to be highly non-linear in its resistance to deflation. They concluded that the non-linear behaviour was a reflection of the folding around the adradial joints, as described by Gladfelter (1972). They suggested that the compliant low strain region corresponded to the folding stage, while die stiffer higher strain region of the pressure-volume behaviour was due to the action of the radial fibres. Earlier in this thesis, I measured the stiffness of the radial fibres and die matrices of the joint and bell mesoglea, then integrated them into an engineering model of the jellyfish which correctly predicted its pressure-volume behaviour, thereby demonstrating tiiat the joint and fibres model proposed by DeMont & Gosline (1988a) could in fact account for the observed behaviour. I have also measured the hyperinflation side of the pressure-volume loop, where die animal is being stretched circumferentially. Experiments up to tiiis point have been quasi-static. I now turn my attention to the dynamic beliaviour of die jellyfish bell. DeMont & Gosline (1988c) converted the pressure-volume data from their experimental set-up into circumferential stress and strain using a tiiin-walled cylinder model, then fit a linear oscillator model to their observations. They approximated the non-linear stress-strain beliaviour of the jellyfish with a linear Hookean model of stiffness equal to the slope of the stress-strain curve at 5% deflation. They measured the damping from a free-swimming animal, estimated the mass of the system and concluded that the animal was likely resonating at its swimming frequency near 1Hz. In tiiis chapter, I take another look at the concept of resonance in a non-linear system. I conclude, like DeMont & Gosline (1988c), tiiat the animal is likely gaining the benefit of resonance - an energy savings of 30% or greater - and find that the form of that resonance is very different. I show also tiiat Gladfelter's 72 (1972) observation of multiple swimming frequencies can be explained through die non-linearity of the system. I find that die mounted jellyfish preparation first described by DeMont & Gosline (1988a) resonates at a frequency much higher dian the swimming frequency of Polyorchis, and conclude tiiat the mounting process affects both die stiffness of die mesogleal bell and the effective mass of the jellyfish oscillator. I conclude the chapter witii a model which compensates for die effect of mounting to predict the elastic behaviour of free-swimming jellyfish of all sizes. The Model The theory of oscillating systems has long been understood, particularly for ones with linear spring restoring forces (Hooke 1678). However when the restoring force is non-linear, and cannot be approximated with a linear function, the analysis becomes more complicated (French 1971). For small deviations from the linear case, there is an analytical approximation (Zeeman 2000), but for larger deviations, die investigator either lias to accept the limitations of a linear approximation, or turn to numerical methods. One characteristic of an oscillator is tiiat it has a natural, or resonant, frequency, whose value is a function of the stiffness of the spring and the mass of the system, at which the system will oscillate if left alone after an initial disturbance. If friction is present in the system, the amplitude of oscillation will decrease witii time, and die resonant frequency will be shifted downward by an amount in some way proportional to the magnitude of the frictional force, or damping. The resonance occurs because at tiiat frequency, the energy cost of motion is least, since the energy stored in the spring exactly equals die energy required to stop and restart the motion of the system at die end of each oscillation. The phenomenon manifests itself in a plot of amplitude or force against frequency. The amplitude created by an external force of given magnitude but varying frequency will be greatest at the resonant frequency. Alternatively, the force required to cause die system to move through a given amplitude will be least at die resonant frequency. This is shown graphically in Figure 5.1. The advantage to die animal of operating at the resonant frequency is a decreased cost of locomotion. The shape of the trough in Figure 5.1 depends on the magnitude of die damping force, but die relative magnitude at resonance can be quite significant. DeMont & Gosline (1988c) used energy conservation arguments to suggest a possible energy savings of 30% for Polyorchis. I will show later in tiiis chapter tiiat diis is correct for large animals, and tliat die potential energy savings are even greater for smaller ones. 73 Lissajous Figures - Linear Oscillator 0.1Hz 1.0Hz 2.2Hz 4.0Hz Extension (mm) Resonance Curves - Linear Oscillator CD 3 0.018 0.016 0.014 0.012 0.010 i 0.008 O 0.006 L L 0.004 0.002 0.000 Low amplitude High amplitude B 1 2 3 4 Driving frequency (Hz) Figure 5.1. Resonance behaviour of a linear harmonic oscillator. A. Force-extension behaviour of a linear oscillator as the driving frequency increases. Vertical and horizontal scales are the same in all panels. The left-most panel shows the quasi-static case, where the force required to generate a displacement is in exact phase with the displacement. As the driving frequency increases, the loop opens as the velocity-dependent damping term begins to take effect. The third panel shows the resonant condition: the force and displacement are exactly 90° out of phase, and the force amplitude is at a minimum. As frequency increases beyond the resonant frequency, the force and displacement fall further out of phase and the force amplitude rises again. B. Resonance curves of a linear oscillator forced at differing amplitudes. As the driving frequency increases from zero, the force required to achieve a given amplitude of oscillation decreases until the resonant frequency, then increases again. The resonant frequency is independent of the amplitude of oscillation. 74 Resonance in a simple harmonic oscillator The equation of motion for a harmonic oscillator can be written as: mr = F - F - F (5 1) IIIJK, i driving spring damping  v ' ' where m is the total mass of the system and X is die acceleration of the centre of diat mass. Each of the force components can be represented as a function, as follows. In a simple harmonic oscillator, the spring restoring force, F s p ring, is assumed to be a linear function of extension (x) with constant of proportionality k. Damping, defined as the loss of energy from die system due to frictional forces, is assumed to be a linear function of the velocity of the centre of mass, with proportionality constant b. The driving function is usually modelled as a sinusoidal function of time, witii amplitude F, and circular frequency co. After substitution and rearrangement, the equation of motion becomes: mx + bx + kx- F0 cos(a>/) (5.2) The solution of this differential equation is well known (French 1971). Defining co0=k/m and y=b/m: x(t) = B cos(cot - 8) (5.3) where: F0(o)) = Bm^{(ol-(o2)2 +{/(o)2 (5.4) and: tan S(eo) = 2r(° 2 (5.5) (OQ - CO The phenomenon of resonance is shown graphically in Figure 5.1. The top four panels are Lissajous figures (force-displacement) which show the dynamic behaviour of a driven harmonic oscillator. In die left panel (low frequency), die force and displacement are in pliase, and die system follows die spring function very closely. As the frequency increases, the velocityrdependent damping begins to have an effect, and the loop begins to open. The third panel shows the phenomenon of resonance - the force and displacement are now 90° out of phase, and die force amplitude is at a minimum. At high frequency (right panel), die force and displacement are close to exactly out of phase, so die force to create the oscillation lias increased dramatically. This is summarised in die lower panel, which shows force amplitude as a function of 75 frequency for increasing amplitudes of oscillation. The frequency where force is a minimum is die resonant frequency. This is also the frequency at which the system will oscillate if left alone after an impulse. Note that die resonant frequency is independent of oscillation amplitude. Non-linear oscillator model In non-linear systems which operate over very small strain ranges such diat die stiffness of the system can be approximated by the local slope, the linear harmonic oscillator model is sufficient to explain the resonant behaviour of die system. However, when die strain ranges are large, as is die case with die jellyfish, it no longer suffices, and a more complex model is required. If die spring restoring force can be modelled as a linear function with a cubic correction, tiien there is an approximate solution (Duffing 1918) which has the characteristic of its resonant frequency being a function of the driving amplitude (Figure 5.2). If die system is highly non-linear, die resonant frequency is an unstable state, and the amplitude at resonance is impossible to predict (Zeeman 2000). The dashed line in Figure 5.2 shows the unstable solution of die equation - there are at least tiiree solutions to the equation in the resonant condition. At these frequencies, with the right combination of parameters, the system tends to go chaotic. Since there is no analytical solution to the general non-linear oscillator equation, numerical methods have to be used. The model development proceeds in a fasliion identical to tiiat used in die linear case. Defining m as die total mass of the system, tiiat is the mass of die jellyfish and the affected water around it, and x as the acceleration of the system, the forces acting on the jellyfish can again be collected into the balanced force equation: ™X — Faring ~ Fspring ~ Fjamping (5.6) The departure from the linear model conies in die definition of the force components. Functional forms for these are developed below. Spring restoring force For die purposes of the analysis that follows, what is required is a matiiematical representation of the force which opposes the movement of water in and out of the subumbrellar cavity of the jellyfish. Physically, the resistance to volume change is provided by wall stresses in die bell, more specifically, die resistance to deflation is provided primarily by the stretch of radially oriented fibres in die bell wall, and secondarily by circumferential and radial stresses on the mesogleal matrix itself. DeMont & Gosline (1988a) modelled the system as a tiiin-walled cylinder, effectively representing the tensile stress in the radial direction as a compressive stress in die circumferential direction. Their effective compressive stiffness also includes the axial stress due to the capped boundary conditions at each end of the cylinder and therefore overestimates the stiffness of the material. However, for the purposes of predicting die resonant behaviour of the mounted 76 \ Catastrophic drop in amplitude Figure 5.2. Resonance behaviour of a non-linear harmonic oscillator (Duffing's equation). The figure shows displacement amplitude, A, as a function of driving frequency, co. The oscillator shown is being driven by a constant force, rather dian die constant amplitude shown in Figures 5.1 and 5.3. The displacement amplitude increases with driving frequency as it would in a linear case, however, at the resonant frequency, die peak curls over and in any physical system there is a catastrophic drop in amplitude. The dashed line shows the unstable state of die oscillator near die resonant condition. 77 animal, their model will suffice, with die effective total hoop stress (CTc) of die structure defined as follows: Pr a c = — (5.7) T where r is the instantaneous internal radius, x die wall tiiickness, and P the internal pressure8. The area over which the stress acts is the longitudinal sectional area, namely: A„ = 2Ths (5.8) where h s is die shoulder height of die mounted bell. The wall tension is therefore: T = acAls=2P(t)r(t)hs (5.9) where hs is as above and r is the resting radius of the animal. Pressure measurements made as described below were converted using die above formula, while r(t) was obtained from die volume by: Once the axes had been converted, die functional form of die spring function was obtained from a third order polynomial fit to die resulting curve, i.e.: 1=0 The exact form of the spring function is not important, only tliat it faidifully represent the shape of die force-extension plot over die range of extensions being studied. Damping The damping force was modelled here as a linear function of velocity, namely: Fdamping=b.f (5.12) where die dot indicates differentiation with respect to time. The value of the damping coefficient has to be determined experimentally. DeMont & Gosline (1988c) observed a transient oscillation ofa 8 Note tiiat as in Chapter 4, this is the pressure "behind" die muscle (ie., on die mesogleal side of the swimming muscle), ratiier than the pressure measured inside the subumbrellar cavity. 78 tethered, but otherwise free-swimming jellyfish, tiien calculated a damping coefficient using die logarithmic decrement method. However die value could not be used here (though it will be used later in the discussion of the free-swimming case) because the fluid flow conditions near the mounted jellyfish were very different from those surrounding die free-swimming animal. Some damping arises due to friction within the mesoglea, but most of the damping in die free-swimming case results from energy losses associated with die movement of water around die bell, whether in the trailing edge vortices, or in the wake, while in the mounted case, die Plexiglas prevents die circulation of water, and, because the animal is not moving forward, it leaves no wake. Thus die damping coefficient for die mounted case is expected to be substantially lower tiian in the free swimming case. In any event, it must be determined experimentally, measured while die animal is mounted. This was accomplished by tuning die model parameters to the experimentally observed behaviour of the mounted jellyfish preparation, as will be described in a later section. Mass Because the fluid surrounding die jellyfish is of a similar density to the mesoglea, it has to be taken into account in the mass of the oscillating system. There are therefore three components to the total mass: the mass of the bell wall, the mass of the slug of water being driven by the piston, and the mass of water outside the bell, entrained by the motion of the bell wall. The apex was not included in die oscillating mass since it was constrained not lo move (see methods, below). The mass of die bell wall was calculated as follows: where p is die density of the mesoglea, assumed here to be effectively equal to tiiat of sea-water, hs is the shoulder height, x tlie resting wall thickness, and r the resting radius of die animal. Because the upper and lower ends of the bell wall were also constrained by the apex and Plexiglas, respectively, only the middle section was able to move. An adjustable parameter, a, was introduced into the mass model to take tiiis into account. The mass of the slug of water was assumed to be that of die largest volume of water removed from die system by the piston. Because there was no cavitation outside die bell during these experiments, nor did the fluid heat up, the principle of conservation of mass (Lightliill 1986) was assumed to apply, and tlie mass of the entrained water was taken to be equal to the mass of the slug. Tlie total mass was therefore modelled as: (5.13) (5.14) Tlie value of a was determined from the best fit of the model to the experimental data. 79 Driving Force & Solution Once all of the functional forms have been substituted into Equation 5.1, tlie model becomes: mr + br + ^ atr' = F(co,t) (5.15) Both tlie model and die mounted jellyfish were displacement-driven, which greatly simplified both the computational algoritiim and die experimental protocol. Tlie PID-controlled actuator (described below) ensured tiiat tlie oscillations were perfectly sinusoidal, and at a designated amplitude, at all frequencies. Numerically, r(t) was set equal to B sin (cot), where co was the driving frequency in radians/sec, and B was the amplitude applied in the corresponding experiment. Tlie function was tiien differentiated twice, values substituted into Equation 5.15, and die model allowed to run for several cycles. Tlie solution to die model is shown in Figure 5.3. The panels are equivalent to those presented for the linear model in Figure 5.1. At low frequencies, the system follows the spring curve. As tlie frequency increases, tlie loop opens due to die effect of the damping function. Resonance arrives in two phases. At first die low-stiffness plateau region goes horizontal as it goes through its resonance, tiien tlie stiffer higher-strain regions go tiirough their resonances at higher frequencies. Tlie overall resonance is defined as the frequency at which the total force required to create a given oscillation amplitude is least. At very high frequencies, die loop flips over. Tlie model makes die additional prediction tiiat tlie resonant frequency should increase with increasing oscillation amplitude (lower panel, Figure 5.3). This would allow the jellyfish to adjust its swimming frequency and still take advantage of the energy savings of resonance. Note die unstable solutions to Duffing's equation (dashed line in Figure 5.2) do not appear since tiiey are not relevant to tlie jellyfish (except possibly in some ineffectual contractions tiiat occur when tlie animal is ill, damaged, or coming out of anaediesia and unable to control its contractions. Although mathematically interesting, tiiis situation only very rarely occurs in wild jellyfish, so I have not pursued die matter further.) Materials & Methods Study animals Animals for tiiis study were obtained from the waters of Esquimau Harbour, near Victoria, BC. They were maintained in a recirculating seawater aquarium at 11°C for up to 2 weeks before use. Jellyfish were fed live Artemia (3-5mm) every other day, and only healthy, actively swimming animals were used in experiments. All experiments were conducted in seawater, maintained at 11°C. 80 Lissajous Figures - Non-Linear Oscillator z \ \ 4.0Hz 0.1Hz 0.5Hz 2.2Hz Extension (mm) Resonance Curves - Non-Linear Oscillator 0.020 0.018 0.016 0.014 •8 0.012 | - 0.010 re g 0.008 Li- 0.006 0.004 -| 0.002 0.000 Low amplitude — — «• Med amplitude * •« • •«« High amplitude -r -4 0 1 2 3 Driving frequency (Hz) Figure 5.3. Solution to the non-linear oscillator. A. Lissajous figures. As before, the axes are identical in all panels. At low frequency, the oscillation follows the spring function. As the frequency increases, the loop opens up due to the damping. The resonant condition is shown in the third panel. At very high frequencies, the loop flips over and closes up again. B. Resonance curves. The resonant frequency in the non-linear oscillator is a function of driving amplitude. 81 Apparatus Tlie experimental set-up was similar to tlie one described by DeMont & Gosline (1988c), and described in detail in the previous chapter. Jellyfish were glued by Uieir margins with cyanoacrylate adhesive to a sheet of Plexiglas. A hole bored in die Plexiglas allowed a support stem made of 0.6cm brass pipe to be inserted into the jellyfish bell cavity. The inserted end of tlie brass pipe was capped, and a set of small holes was bored into its side wall. A collar was soldered to the pipe to ensure that it could not be inserted too far. A machine screw tiireaded into die end cap supported the peduncle of the jellyfish such that the animal could not crumple longitudinally. The open end of the pipe was connected through flexible plastic tubing to a piston driven by a PID-controlled Ling Systems electrodynamic actuator. A stainless steel cadieter was inserted dirough die brass tube, filed flush widi die side wall, and silver-soldered in place. Tlie catheter was connected to a variable reluctance pressure transducer (Validyne model DP103, membrane 8-24) which was assembled underwater to ensure there was no air in the system. The apparatus was controlled and monitored using software custom written in die Lab View programming language. Calibration Careful dynamic calibration was necessary since die system compliance of the apparatus and pressure transducer was such that its resonant frequency was very near to tiiat of the jellyfish. With a high-quality calibration, it was possible to extract useful data from die experimental apparatus using a deconvolution algorithm, as discussed by Kasapi & Gosline (1996). Volume Tlie linear displacement of die Ling actuator arm was converted to a volume change as described in the previous chapter by multiplying the displacement of tlie piston head by the internal cross-sectional area of the piston. To account for die phase shift and amplitude loss observed in die plumbing between the piston head and die jellyfish, tlie system was calibrated dynamically as follows. A graduated pipette and plastic sleeve assembly was placed over tlie support stem at die end of the usual plumbing. A solution of red food colouring and dish soap was added to the top of the water column in tlie pipette to minimise tlie size of the meniscus. Tlie actuator was ramped tiirough a frequency range of 0.1 to 6 Hz. Tlie motion of the meniscus was followed using a video camera and video dimension analyser (VDA: Model 303, Instruments for Physiology & Medicine, San Diego, CA), which tracked the intensity contrast boundary in die video image. The resulting complex-valued volume transfer function is shown in Figure 5.4. The voltage signal output by the strain gauge built into tlie Ling actuator was converted to volume change at the jellyfish through complex multiplication by die transfer function. Pressure Ideally, one selects a transducer with a resonant frequency well above the frequency range to be tested, such tiiat the transducer's resonant behaviour does not affect die data. However, because the pressure 82 Displacement Transfer Function Driving frequency (Hz) Figure 5.4. Transfer function: Piston displacement to bell cavity volume change. The system was characterised out to 6Hz and a linear oscillator model with resonant frequency 6.5Hz and quality, Q = 6.0, fit to the calibration data. 83 changes being measured in die jellyfish bell cavity were very small, a highly compliant transducer was required. According to die manufacturer, die variable reluctance transducer (Validyne DP-103) used here could measure full-scale pressures as low as 1 mm of water using its most compliant membrane. This had die unfortunate consequence of compromising die transducer's ability to follow die signal at frequencies above 1 Hz. Nevertheless, much of the higher frequency information could be recovered by deconvoluting the transducer transfer function from die pressure trace. The transducer was made of two coils separated by a stainless steel membrane. The coils were set into a pair of stainless steel plates with an indentation machined onto one side. When die transducer was assembled, the membrane created a cavity between itself and the indented side of each plate. O-rings ensured a water-tight seal between the plates and die membrane, and a tiireaded hole tiirough each plate allowed water to flow in and out of the cavities, whose volume was about 57 uL. In the experiments reported here, die transducer was assembled underwater, with care taken to ensure tiiat no air bubbles were trapped inside of it during assembly. Electrically, tlie transducer is best described as an inductive half-bridge (Validyne 2002). The coils were energised by an alternating current carrier signal. As the membrane was forced nearer to one or die other coil by the motion of water, tlie change in tlie inductance of the coils brought die half-bridge out of balance, and an AC signal resulted. This was demodulated and transmitted to the recording computer as a DC voltage. Tlie magnitude of the change in inductance was linearly proportional to die displacement of die membrane, and hence to die applied pressure. Static calibration of die pressure transducer was done using a manometer. To measure die dynamic transfer function of the transducer and its associated plumbing, die following protocol was adopted. The transducer was connected using its usual plumbing to a 50 mL Nalgene bottle by inserting tlie catheter tip through the side of the plastic bottle, securing it in place with epoxy. Tlie bottle was filled about half way (well above the catheter insertion point) with water and its top was sealed with a rubber stopper. A glass tube inserted through die rubber stopper was connected to a 1 L vacuum flask, itself connected to a piston driven by the same PID-controlled actuator used throughout tiiis thesis. The motion of the piston created a wave of compressed air, which travelled tiirough the plumbing to tlie Nalgene bottle, forcing tlie meniscus up and down. This small pressure change was tiien transmitted tiirough tlie water to tlie transducer. The transfer function for tlie transducer and its 8-24 membrane is shown in Figure 5.5. The function looks qualitatively like a linear oscillator, but no combination of parameters will make it fit. Tlie linear oscillator phase function (Equation 5.5) was fit to the pliase data, but a polynomial function was required to fit die magnitude data. Curves were fit to die data using TableCurve (Jandel Scientific). 84 Pressure Transducer Transfer Function 3.0 0 1 2 3 4 5 6 7 Driving frequency (Hz) Figure 5.5. Transfer function: Pressure transducer to bell cavity pressure. The system was characterised out to 6Hz. It was not possible to fit a linear oscillator model to the amplitude function, so a polynomial function was used instead NA = (1.0593-0.035co+4.73xlO"4co2)/(l-0.091co+2.68xlO"3o2). The phase function could be described by the phase component of a linear oscillator model (Equation 5.5) with resonant frequency co0 = 2.6Hz, and y = 12. 85 Deconvolution algorithm Tlie effectiveness of the deconvolution algoritiun used here was verified experimentally by Kasapi & Gosline (1996). The signal was first multiplied by its static calibration constant. It was next input to a fast Fourier transform subroutine (Press et al. 1992). Tlie resulting complex function was broken into magnitude and phase components. Tlie magnitude of die transformed signal was divided, frequency by frequency, by tlie magnitude component of the transfer function. Likewise for tlie phase signal, except that the transfer function phase signal was subtracted from die transformed data. The corrected signal was then back-transformed using the FFT algorithm, then normalised by dividing by tlie size of the input array. The deconvolution method is reliable up to a certain point. Once the amplitude gets too much reduced by the compliance of the transducer and the frequency roll-off of the filter built into its electronics, attempts to rescue die information just increase the noise to unacceptable levels. For tiiis reason, I limited die deconvolution algorithm to 3 Hz, even though I had cliaracterised die transfer function out to 9 Hz. Testing protocol Once the measurement apparatus had been calibrated, die quasi-static behaviour of the preparation was recorded at 0.1 Hz. Next, die computer cycled the system through multiple frequencies, from 0.1 Hz to 3.5 Hz, in random order, at an amplitude similar to tiiat observed in free-swimming animals (Figure 2.4). The actuator was controlled by a PID algoritiim (National Instruments), tuned such tiiat the piston oscillated sinusoidally at all frequencies. Results Spring restoring force Tlie pressure-volume data presented in the previous chapter were converted to stress and strain, and thence to circumferential force and extension as described in Equations 5.7-5.11. Seven of the nine jellyfish tested in Chapter 4 were selected for dynamic testing in this chapter: force-extension data are presented in Figure 5.6. Each colour in the figure corresponds to an individual jellyfish, while the dashed or solid lines show different amplitude experiments. Tlie tiiick solid line is die global average, computed as described in die previous chapter. Third-order polynomials were fit to the data - coefficients are presented in Table 5.1. The non-linear shape of the curves is evident. In deflation, there is a low-stiffness plateau, as the bell first folds around die adradial joints. In higher strain regions - typically beyond about 10% - die stiffness increases abruptly as tlie radial fibres are recruited. In inflation, die stiffness rises immediately as the muscle and skins are stretched. The average local stiffness in the joint region (measured at 1% deflation) was 140Pa, while at higher strain (12% deflation), die average local stiffness was 607Pa. 86 Force - extension, Mounted Jellyfish Preparation Figure 5.6. Force-extension behaviour of mounted jellyfish. Converted from pressure-volume data assuming a capped cylinder model. Colours correspond to individual animals; dashed or solid lines refer to different experiments. Thick solid line is global average. Fit parameters and local slopes (stiffnesses) are summarised in Table 5.1. 87 Jelly ID & test # Fit parameters Local Stiffness (Pa) ai a 2 a 3 Joint Deflation Inflation a 8 0.196 58.19 30696 57.34 664.58 458.80 c 35 0.281 23.88 16545 80.58 380.62 250.60 d 47 0.337 50.45 26650 115.88 603.62 451.69 d 66 0.333 41.85 21093 116.22 496.08 386.49 e 27 0.423 33.13 26709 170.37 683.25 459.03 e 57 0.260 33.00 20060 101.54 459.9 337.88 f 51 0.148 17.83 4931 44.34 74.62 97.70 f 68 0.168 16.99 4384 50.86 75.19 100.20 g 32 0.330 33.51 24638 113.08 496.87 341.67 h 16 0.951 43.80 44436 314.68 1670.24 936.59 h 17 1.005 50.25 51020 332.13 1888.67 1046.08 i 15 0.257 15.20 9388 84.20 240.57 182.65 k 9 0.365 24.36 10606 164.07 308.62 299.23 k 57 0.472 43.92 18266 208.96 451.34 446.51 Global avg. 0.395 34.74 22102 139.59 606.73 413.94 DeMont 400-1000 Table 5.1. Third order polynomial fits to the force-extension curves. Note that since the data were normalised such tiiat tlie inflection point in the function was defined to be zero pressure and volume, die zero-order coefficient, ao, was set to zero. Also shown in the table is the local deflation stiffnesses in tlie joint (1% deflation) and high (12% deflation) strain regions, and the local stiffness at 5% inflation strain. Tlie 12% strain was selected to enable comparison of the deflation stiffness measured here with stiffness data presented by DeMont & Gosline (1988a). 88 Kinematics, mass & damping With proper selection (described below) of system mass and damping coefficient parameters, the non-linear oscillator model very closely predicts die dynamic behaviour of the mounted jellyfish preparation, for frequencies up to about 2 Hz (Figure 5.7). At low frequency, die Lissajous figure is completely closed, and follows the polynomial spring force function described above. As the driving frequency increases, the loop begins to open as die effect of tlie velocity-dependent damping term becomes significant. As the resonant frequency is approached, the loop turns sigmoidal in shape as the low-stiffness plateau goes through its resonance before the higher-stiffness tail regions do. Tlie overall resonant condition is achieved in the 2.1Hz panel of Figure 5.7 A, where the total force amplitude is lowest. As frequency is increased even further, the higher stiffness tail region goes through its resonance and the loop flips over. The jellyfish and transducer were too compliant, even despite the deconvolution algorithm, to show the complex behaviour at very high frequencies. Tlie quality of the model fit at low frequencies was similar in all jellyfish tested, confirming that the physics of the non-linear oscillator model is correct, and tiiat it can be used to predict die behaviour and resonant frequency of the mounted preparation. Tlie system mass and damping coefficient were determined empirically as follows. First, the damping coefficient was set to zero (which removed the hysteresis), and die mass was adjusted until the model prediction was parallel to the data for all frequencies. Next the damping coefficient was increased until the hysteresis loop opened to the same extent in tlie model prediction as was observed in the data. Finally, both parameters were tuned until tlie best fit was obtained. Values for the damping coefficient and total mass are presented in Table 5.2 for each of the jellyfish tested. Tlie damping was very small, averaging about O.OlNs/m, and die mass averaged about 2.9g. Note tiiat both values are very much smaller than in the free-swimming case (Table 2.2), where tlie value of the damping coefficient was estimated to be about 0.04Ns/m and die mass about 15g. Tlie best fit value of the adjustable mass parameter, a, which was introduced in Equation 5.14 to take into account tlie restraint on tlie bell wall of tlie peduncle and margin, was about 0.74. This seems quite reasonable given the geometry of the preparation, particularly the variable stiffness of tlie peduncle and shoulder joint, and die variable penetration of tlie cyanoacrylate into the base of the bell wall. Resonant frequencies Once the mass and damping parameters were determined, die model could be used to predict die resonant frequency of die system. Figure 5.8 shows the match between model & experiment of force amplitude as a function of driving frequency for a typical experiment. Beyond 2.7 Hz, die experimental apparatus and the jellyfish itself were no longer able to respond meaningfully to tlie driving function. Resonant frequencies for all jellyfish are presented with tlie mass and damping parameters in Table 5.2. 8 9 Figure 5.7. Kinematics of two typical experiments. Grey boxes are experimental data. Solid red line is the non-linear oscillator model prediction. Model circumferential amplitude was selected to match the experimental amplitude. Mass and damping coefficients were adjusted until die best fit (determined by inspection) was obtained. The model very closely predicts the behaviour of die experiment until about 2 Hz, after which diere is no longer enough information available in die experimental data due to the compliance of the pressure transducer and die jellyfish itself. Vertical axes are hoop tension in N, horizontal axes are circumferential extension in m. Suffices'm' and 'u' denote 10"3 and 10"6, respectively. Animal dimensions, mass and damping coefficients are given in Table 5.2. Spring function coefficients are presented in Table 5.1. 60 l3 r S • . <N © w> r o ©' rs — u o r s r s *s VO r S I™ ; ID •S j v © r o oc r- r s ' t o <3v V , CO C <n' oo' v© r o J O i O • r s 5 © j r s : V r s r s v£ ID r O © r s r** r s «D © CO r o r o r s « i 00 © 00 r s r o r s j c o ro 1 - "2 M . -rs va ^ ^ ©' © r s r s vfi © </-. 00 O rs ro* «-J - t rs mm IH1 o r s ' r-o im l A Iv (-1 v . 00 ©' o C N © r o * * ©' Ov - D r s r s Os C rs rs soo:* © i n CO ! C O : a 2 r s t v C r o © ro' r s 3 V vC r s i r o © r - wo r o ri-" r s © r o r s r S © rs 00 r s r s ' r o r s ' to is rs ^ \o VO 00 © r s \0 . — i r o © m r s - T oc r» ©' ~i —' rs ro Ov r--r o Q : I/) € iii: v i m o "5b "5b ! CJ ; Cu s 2 s i t s 8 -8 "3s l i l i i i i i i i s 60 e 'S, S re a 00 <3 " ? 1 rsi •d 2 c r TJ ~ OH G O 93 Resonance curve (Jellyfish E) 0.0035 0.0030 0.0025 H 9r 0.0020 O 0.0015 0.0010 0.0005 0.0 0.5 1.0 1.5 2.0 2.5 Driving frequency (Hz) 3.5 Figure 5.8. Experimental verification of nonlinear resonance. Data points are tlie maximum force amplitudes from die experiment reported in Figure 5.7 (Jellyfish E). Solid line is die maximum force amplitude predicted by the non-linear model. 94 Tlie average resonant frequency for tlie seven animals tested was 2.26 Hz. This is more than double die swimming frequency observed in Cliapter 2 for animals of equivalent size. Discussion Non-linear springs are quite common in biology. Tendons, arteries (Fung 1993, Shadwick & Gosline 1985, McConnell et al. 1997), ligaments, blubber (Pabst 1999), squid mande (Gosline & Shadwick 1983a), and jellyfish mesoglea (DeMont & Gosline 1988a) are examples of tissues which exhibit j-shaped stress-strain behaviour, reflecting die underlying structure. In many cases, die non-linearity is not really evident in vivo, since the system operates over a small strain range at a given pre-strain. However, in other systems such as die jellyfish, which oscillate over large strain amplitudes, the non-linearity can not be ignored. Tlie model presented here is able to predict the resonant frequency of the mounted jellyfish system. It does so with an empirical measurement of die damping coefficient and a theoretical estimate of the mass of the fluid involved in the oscillation, both of which are subsequently confirmed experimentally. With a realistic physical model of die system and its boundary conditions, a non-linear spring function was derived from a quasi-static experiment. With all of the above information incorporated into the generalised harmonic oscillator equation, the appropriate driving force could be applied to predict die experimentally observed resonant beliaviour of die mounted jellyfish preparation. The resonant frequency of tlie preparation was found to be approximately 2.2Hz, about twice the observed oscillation frequency of similarly sized free-swimming jellyfish (Chapter 2). This apparent paradox can be resolved on closer inspection of tlie underlying physics. In die free-swimming case, as discussed next, die mass of fluid involved in the oscillation is very different, indeed it is some 5 to 6 times greater. Tlie mass is not die whole story, however. The boundary conditions on the jellyfish mesoglea are obviously very different. Tlie effect of gluing the jellyfish to tlie Plexiglas, then supporting die peduncle on the support stem was to stiffen (lie bell relative to tlie free-swimming case. The free-swimming animal It is impossible to measure the stiffness of the bell during free-swimming, because to do so would interfere - indeed destroy - die animal under study. However, it is possible to make predictions based on the model of the behaviour of the whole system, using the material properties measured in Cliapter 3, combined with tlie tiiick-walled cylinder model of jellyfish geometry presented in Chapter 4, and the non-linear harmonic oscillator model developed in tiiis Cliapter. Making a few assumptions about how jellyfish dimensions scale with size, it is further possible to make predictions about die locomotor behaviour or jellyfish of all sizes, which can be compared to data presented in Chapter 2 to test the validity of the model. 95 Spring function The model of die spring function begins with die assumption diat the material properties of the mesoglea (matrix and fibres) are independent of body size. Tlie second assumption is that die cridcal strain parameter, sc, which relates the circumferential length of the gastrodermal lamella to that of the subumbrellar epidielium, is invariant with size. I do not have enough data to prove either, but for the purposes of generating a hypodiesis, tiiey both seem reasonable to assume. In Chapter 2,1 showed that the outer dimensions of the animal scale geometrically with bell height. In Chapter 4,1 derived a functional form to predict die cavity pressure in the mounted jellyfish preparation. That derivation was based on Roark's formula (Young 1989) for a thick-walled cylinder, capped at both ends. Tlie removal of the boundary condidon at the margin end of the jellyfish converts the closed cylinder into an open cylinder (Wright 2001). Tlie difference between die two condidons is tiiat in Equation 4.8 goes to zero in tlie open case. Tlie open cylinder equivalent to Equation 4.14 tiien becomes: P = £90^C 2 r lr.'-r.') F F — r2{\ + v)+r2{\-v) / 2 2 X re2(\ + vTL) + rt2(\-vTL)\ (5.16) where all symbols are as defined in Chapter 4. As before (Equation 4.13), since the volume of tlie bell mesoglea does not change on a time scale relevant to locomotion, re can be expressed as a function of r„ 2 2 , r. = r_. + 7th, (5.17) In order to make use of the non-linear resonance model developed earlier in this cliapter, I introduce a new effective stiffness, and convert pressure to circumferential tension using tlie thin-walled cylinder model: F(t) = 2P(t)r(t)hs (5.18) where P(t) is as defined in Equation 5.16, h, is tlie shoulder height of the jellyfish, and as before, r(t) is derived from die cavity volume change, AV(t), by: (5.19) 96 where r 0 is the resting bell radius. The combination of Equations 5.16 to 5.19 and die assumptions about scaling made above produces the predicted free-swimming spring restoring force functions shown in Figure 5.9 for jellyfish ranging in bell height from 7 to 40mm. Forcing function The shape of the forcing function was measured by Spencer & Satterlie (1981), and reproduced in die vertical axis of Figure 2.8. Their force transducer was however uncalibrated, so it was not possible to determine the magnitude of the force from tiieir research. DeMont & Gosline (1988c) estimated the magnitude of the forcing function by assuming the stress generation of the jellyfish muscle to be similar to other striated muscle, namely 2.0 x 105 Pa. They used Gladfelter's (1972) measurements of the cross-sectional area of the subumbrellar myoepithelium and calculated tiiat the muscle could produce about 0.0 IN of force. Occasionally some jellyfish would continue to contract throughout the experiment, despite being glued to the Plexiglas. This behaviour rendered die experiment useless for the measurement of the elastic properties of die mesoglea, since die muscle was obviously active and potentially changing die stiffness of the bell. However, such instances did make it possible to verify DeMont & Gosline's (1988c) prediction of muscle force. This behaviour occurred infrequently (due to die injury to the tissue of die nerve rings by die cyanoacrylate during the mounting procedure), and only once during an experiment in which all of die equipment had been properly calibrated. Figure 5.10 shows a typical pressure trace from tiiat experiment. Removal of die underlying volume displacement-induced pressure signal made it possible to measure the amplitude of the pressure pulse. The average pressure generated by the muscle was approximately 14 Pa. Modelling the muscle sheet as a thin-walled cylinder witii inner radius r0=0.0078m, shoulder height hs=0.021m, and die thickness of the sheet as 2um (Gladfelter 1972), die stress generated by the muscle can be estimated at 5.4xl04 Pa. This translates to a circumferential force of 0.0046N. Interestingly, Lin & Spencer (2001) find a similar maximal force value: they measured contractions of approximately 0.4mN in 2-3 mm wide strips of excised velar muscle. Assuming tiiat die swimming muscles perform similarly, this translates to approximately 4mN for jellyfish of 20-30mm shoulder height. To extrapolate from the observed force, die maximum muscle stress was presumed to be size invariant, while die dimensions of the muscle sheet were modelled as scaling geometrically with bell height. System mass The second determinant of the natural frequency is the mass of the system. In the classical harmonic oscillator, die mass is modelled as concentrated at a point at die end ofa massless spring, operating in a vacuum. This approximation works for dense metals vibrating in air, but for die jellyfish, whose density is 97 Deflation Spring Functions -0.18 " -0.20 -I 1 1 , 1 , 1 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 Change in circumference (m) Figure 5.9. Free swimming spring restoring force for jellyfish ranging in size from 7 to 40 mm. Curves were derived using the anisotropic thick-walled cylinder model of Chapter 4 and assumptions made in Chapter 5 about the scaling of body dimensions with bell height. 98 ~ 0.15 «i Time (s) 3-84 Figure 5.10. Muscle force. Spontaneous contractions by a mounted jellyfish. The animal was glued by its margin to a sheet of Plexiglas, but continued to contract throughout the experiment. The upper panel shows the instantaneous volume of water being driven into and out of the bell cavity by the electromagnetic actuator. The lower panel shows the bell cavity pressure as measured by the pressure transducer. The dotted line in the lower panel is a sinusoidal function with frequency identical to the upper trace, and represents a close approximation to the pressure that would have been measured if the animal had not contracted its muscles. The difference between this approximation and the height of the observed pressure peaks was taken as a measure of the force generation capability of jellyfish muscle. This was then converted to hoop tension by assuming a thin-walled cylinder model for the muscle. 99 almost identical to the fluid in which it swims, die approximation no longer suffices. As the bell wall expands radially after a contraction, it displaces the water lying adjacent to the exumbrella, and sucks water into the bell cavity through the velar aperture at the base of the bell. Using the conservation of mass principle, this vorticity can be modelled as a toroidal vortex whose small radius is the shoulder height of the bell and whose large radius is tiiat of the exumbrellar margin Figure 5.11A is a single frame from a 30Hz movie clip of a jellyfish swimming tiirough talcum powder. The vortex is visible on die left hand side of die jellyfish (flecks of talcum powder highlighted by the white dots). The black arrows on die right hand side of the jellyfish show the direction of rotation of the vortex. Figure 5.1 IB is a drawing of the observed structure. The beginnings of similar structures are also visible in Figure 7 of Colin & Costello (2002) for other hydrozoan species. The mass of the toroidal vortex is given by: mv = B-p-xh*-2nrm (5.20) where p is die density of the jellyfish and seawater, h s is die shoulder height, and rm the margin radius of die jellyfish. An adjustable parameter, p\ is introduced in Equation 5.20 because the vortex is not in fact a solid body, but radier one in which the rotational velocity, and hence angular momentum, initially increases, tiien decreases from the centre to the outside. Using the geometric relationships described in Figure 2.1, Equation 5.20 reduces to: mv=3.636ph3 (5.21) where h is die bell length, from apex to margin, and p die density of seawater (here, 1030 kg m*3). Using the same set of geometric relationships, the displaced mass of the jellyfish and subumbrellar cavity is given by: md=0.267ph3 (5.22) Other authors have modelled the entrained mass of jellyfish. DeMont & Gosline (1988c) assumed the jellyfish was resonating, tiien back-calculated a mass of 0.012 - 0.030 kg to fit their linear oscillator model to die observed swimming frequency of 1.1 Hz for a 2.0 cm jellyfish. Daniel (1985) modelled an added mass coefficient for translating ellipsoids and jellyfish models of various fineness ratios. For Polyorchis (FR = h/Rp, = 0.6), he predicts an added mass coefficient of approximately 2.2. In the same format as the previous two equations, tins becomes: ma=0.S54ph3 (5.23) 100 Figure 5.11. Circular vortex model of entrained mass. B. The grey outline shows the jellyfish in its contracted state, at the beginning of the refill phase. The black outline shows the relaxed state of the jellyfish, at the end of the refill stage. The toroidal vortex is shown in dark grey, with arrows showing the direction of flow. The relative lengths of the arrows indicate faster or slower flow, but are not drawn to scale. As the bell reopens, the water is pushed out of the way of the expanding bell wall. Simultaneously, extra volume is created inside the bell cavity, creating a zone of low pressure which sucks the water in through the velar aperture. The toroidal vortex is created through conservation of mass. Jellyfish drawings adapted from Gladfelter (1972). A. The jellyfish in the image is swimming through a sheet of illuminated talcum powder. The edge of a vortex is visible to the left of the animal, traced by the yellow dots. The small radius of the vortex is shown by the white arrow. The margin radius and shoulder height of the jellyfish are shown in black. The model (described below) is shown to the right of the animal. The upper white curved arrows show the motion of the wall of the bell (here almost fully refilled), while the lower arrows show the suction at the velum. The size of the observed vortex on the left matches that of the model (on the right) quite closely. 101 All three models and DeMont & Gosline's (1988c) mass estimates are shown in Figure 5.12. Note tiiat Daniel's (1985) predictions were made in the context of forward motion, and were based on translating balsa wood models of jellyfish. They are lower estimates since they do not account for die radial oscillations of the bell and the resulting toroidal vortices. Two curves are presented for the toroidal vortex model, one with p= 1, and the second with (J=0.55, which was the best fit to the data in Figure 5.13. Damping The damping term in die model gathers together all of the energy losses in tlie system and represents them as a linear function of velocity. These losses are of two types: there are hydrodynamic energy costs associated with moving water, particular in regards to generating useful forward tiirust, and there is friction present in die mesoglea. The hydrodynamic energy loss is due in large part to die contraction of the velum. The velar muscles contract just before the main contraction of the swimming muscles (Spencer 1982), to create a nozzle which directs die jet of water such that thrust production is improved (Gladfelter 1972). This constriction diverts some energy away from the inertia of the fluid into stretching the contracted velar muscles. There is also energy lost to internal friction within the bell. Tlie very small damping measured in die mounted preparation is likely mostly due to this internal friction. Tlie overall value of the free-swimming damping coefficient was measured in Table 2.2 as 0.04Ns/m for a jellyfish 2cm long. To scale tiiis value to jellyfish of all sizes, a specific damping coefficient was defined, y = b/m = 0.04/0.02 = 2.0. Resonance Previous authors have described Polyorchis as swimming at a frequency of about 1Hz. Gladfelter (1972) reported a range of 0.43 to 1.45 Hz, while DeMont & Gosline (1988c) report 1.1 +/- 0.43 Hz. Arkett (1985) showed a frequency range of 0.3 to 2.1 Hz for 17 tethered jellyfish, and stated that the animals tended to swim at similar frequencies in die wild. Tlie refill frequencies reported here in Chapter 2 ranged from 0.9 to 2.3 Hz (1.24 +/- 0.28 Hz), well within tlie range reported by previous authors. Gladfelter (1972) was die first to note die size dependence of the swimming frequency of jellyfish. He presented data on the contraction duration for 16 jellyfish ranging in size from 7 to 40 mm, and noted that the duration followed a square-root function, but did not provide an explanation for the observation. He did not present refill durations, but he did state that refills generally lasted about 1.3x as long as die contractions, an observation very similar to the 1.43x reported in Chapter 2. Arkett (1985) described a similar size dependence of swimming frequency for animals in the wild, and fit an exponential function to tlie data, but did not provide an explanation for tlie trend either. Data from Gladfelter (1972) and Arkett (1985) are combined in Figure 5.13 with tlie data from Figure 2.5 and Figure 2.7. Assuming a simple harmonic oscillator, an average deflation strain of 0.174 (Chapter 2), linear size-dependent damping, and including die size dependence of stiffness, die tiiree mass models can be compared to the data in Figure 5.13. Tlie displaced mass model has die right form, but is obviously not correct. As expected, since it was not designed for this kind of analysis and hence does not take into account die radial 102 Jellyfish Mass Models 0.12 0.10 H 0.08 J2> 0.06 A ro 0.04 A 0.02 A 0.00 H Toroidal Vortex mv = 3.63 ph3 Toroidal Vortex mv = 2.0 ph3 Added mass ma = 0.854 ph3 Displaced mass md = 0.267pti3 10 15 20 Bell height (mm) 25 30 35 Figure 5.12. Mass of entrained circular vortex model as a function of jellyfish bell height. The dashed curve is the mass of water displaced by the jellyfish and the water contained within die bell at rest. The dotted curve is Daniel's (1985) added mass model for forward accelerated motion. The grey solid curve is the mass of die torus given by Equation 5.20, with (3=1. The black solid curve is the toroidal vortex model widi P=0.55, die best fit to die data in Figure 5.13. The short vertical black line is the range of masses proposed by DeMont & Gosline (1988c). 103 3 H N X i c CD =5 c r L L 1 H Size Dependence of Frequency \ • Gladfelter (1972) • Figure 2.5 • 12.5 H2 Video • 30 Hz Video • Arkett (1985) Displacement Added Mass mm mm mm mm Toroidal Vortex Anisotropic model I — 10 20 30 Bell height (mm) 40 50 Figure 5.13. Size dependence of refdl frequency. Data from Chapter 2 are combined with Gladfelter's (1972) and Arkett's (1985) data as described in the text. The harmonic oscillator model (Equation 8) is plotted for the three different mass models described in the Discussion (m ,^ the displaced mass of the jellyfish and contained water; ma, Daniel's (1985) added mass model; and mv, the toroidal vortex model). The first two are obviously not able to predict the frequency behaviour. The toroidal mass / linear spring model is also unsatisfactory, particularly at small size (R2=0.21). The fourth curve, labelled anisotropic model, is the numerical solution to the toroidal vortex mass model (3=0.55, R2=0.67) with the transversely isotropic thick-walled cylinder model developed in Chapter 4. The assumption is made that the mesogleal stiffnesses and critical volume are independent of body size. Values of the parameters were: E = 300Pa, E T = 600Pa, E L = 1500Pa, and V c = 0.83. Scaling relationships described in Chapter 2 were used for the body dimensions, namely \ = 0.8h, r0=0.37h, and T=0.13h. Maximum deflation and inflation strains were set at 0.17 and 0.05, respectively, the average maximum strains observed in free swimming animals (Chapter 2). Specific damping, y, was set to 2.0. The model is unable to predict the behaviour of very small animals: at very small Reynolds numbers, the viscosity of the fluid overcomes the inertia, and resonance is not possible. 104 oscillations of the bell, Daniel's (1985) added mass model underestimates the required system mass. The toroidal model presented in Equation 5.20 comes closer, but no value of the adjustable parameter, p, will produce a perfect fit. The "best" fit requires p=1.5 (i.e. more mass would have to be involved than predicted by the toroidal vortex), and its R 2 value is approximately 0.2. It is possible tiiat the toroidal vortex model underestimates die mass, but the low R 2 value suggests tliat this explanation is unlikely. Furthermore, a (3> 1 is inconsistent witii die observation made in Figure 5.11. The fourth curve in Figure 5.13 is the non-linear model developed above, which combines the transversely isotropic thick-walled cylinder model of Chapter 4 witii the toroidal vortex mass model and measured values of the muscle force and damping functions. The best fit value of the mass parameter p was 0.55, which seems consistent with die fluid dynamics of the toroidal vortex discussed in Figure 5.11, and described by Colin & Costello (2002). The model much more accurately predicts die resonant frequency for jellyfish of all sizes, as indicated by its R 2 = 0.67 (TableCurve 2D, Jandel Scientific). The better fit is particularly obvious for medium sized jellyfish - die linear model misses die point cloud altogether, while die non-linear model nearly bisects die cloud, leaving residuals of similar magnitude above and below the prediction. I suspect tiiat die wide data spread and consequent failure of the model to predict the frequency of very small animals is due to a fundamental chance in die flow regime at very small Reynolds numbers. At low Reynolds numbers, because the inertial forces are so small compared to die viscous forces, die concept of resonance, which depends on fluid inertia, is no longer relevant. Seen anodier way, die very small animals are likely flow limited, that is, they are simply unable to provide enough energy to move the water at the velocity required to oscillate at high frequency. An alternate explanation may be in my assumptions about Arkett's (1985) data: I have assumed that all of the animals he described were swimming in die resonant gait. It is possible, and Gladfelter's (1972) observations of small animals at high frequencies certainly suggest, that the small animals' refill frequencies were much higher, but tiiat they were in fact swimming in the transient mode - at diese high frequencies, without studying the video, it would be very difficult, if not impossible, to tell the difference. Distinguishing between tiiese two explanations will require further investigation with detailed analysis of video recordings of very small animals. The research would likely be very interesting as a study of swimming at low (but not extremely low) Reynolds numbers. The excellent fit of the non-linear model to the observed behaviour of free-swimming jellyfish implies that the model is correct. It is tiierefore possible to make use of the model to calculate die energetic benefit of resonance to the animal. Figure 5.14 shows the resonance curves for jellyfish between 7 and 37mm in bell height. These curves were derived using the toroidal vortex mass model, die global average spring force function, and die free-swimming specific damping coefficient. The savings is a decreasing function of size, with smaller animals gaining a benefit of nearly 70%, while larger animals are limited to about 30%. It is 105 questionable, however, whether the smallest animals are indeed able to take advantage of this potential energy savings, since at die low Reynolds numbers involved, the inertia necessary for resonance to occur is much diminished relative to tlie viscosity of the fluid. The savings of 30-50% over swimming at very low frequencies is however substantial, and should be highly favoured by natural selection. Indeed the neuromuscular system of the jellyfish seems to have evolved to ensure that animals do swim at their resonant frequency, and no faster: the pacemaker in the swimming motor neurones is tuned to fire at about 1Hz for medium sized jellyfish (Spencer 1995), the duration of the action potential increases with size (Spencer & Satterlie 1981), and the muscle refractory period is such that it will not contract more often than the resonant frequency (A.N. Spencer, pers. comm.). It is interesting to consider die penalty (in lost savings) of swimming at frequencies away from the resonant frequency. Tlie resonance peak is much broader for smaller animals, which might mitigate die loss due to die Reynolds niunber problem. For larger animals, however, die penalty for swimming off resonance is applied very quickly. The amplitude of the resonance curve is equal to Q = coo/y for Q > 5 (French 1971). Therefore since y is assumed to be constant, the maximal energy savings increases as size decreases, but the peak also gets narrower, since die resonant frequency is decreasing, as shown in Figure 5.14. The penalty for operating away from resonance is larger and applied sooner (in terms of Af) for larger animals than for smaller ones. This probably explains die narrowing of the range of frequencies with bell height observed in Figure 5.13 - larger animals are observed to deviate much less from the resonant frequency than smaller ones. Conclusion Tlie rhythmic contractions of a swimming jellyfish are familiar to any observer of marine life. That same observer has also noticed tiiat smaller animals beat faster tiian larger ones, and possibly that individuals do not always beat at die same frequency. The underlying mechanics are straightforward, and die beliaviour of die system is suggestive of a harmonic oscillator. Once the fluid dynamics are understood such tiiat the mass of the system can be determined correctly, it is almost possible to explain die observed behaviour of swimming jellyfish with a linearly damped harmonic oscillator model. However, to predict die behaviour of the system properly, particularly its size dependence, it is necessary to take into account the non-linearity of the system. With an experimentally determined functional form for the spring restoring force, a much better fit to die observed data is obtained. 106 Benefit of Resonance 0.012 0.010 0.008 H Q- 0.006 o 0.004 H 0.002 H 0.000 \ \ / J ! Bell height — i — 0.0 0.5 — i — 1.5 7 mm 15 mm 22 mm 30 mm 37 mm lxl0"4N 1.0 1.5 2.0 Driving frequency (Hz) 2.5 3.0 3.5 Figure 5.14. Benefit of resonance. Shown are predicted resonance plots for jellyfish from 7 to 37 mm bell height Curves were derived using die transversely isotropic thick-walled cylinder model from Figure 4.5, the toroidal vortex mass model with (3=2.5, and specific damping coefficient y=2.0. The benefit of resonance can be derived from die ratio of the force required at die resonant frequency to tliat required as frequency ->• zero. The benefit is a decreasing function of size, with small animals potentially gaining a savings of up to 70%, while larger animals are limited to approximately 30%. 107 Chapter 6. General Discussion The aim of this thesis is to understand die role played by the mesogleal skeleton of the jellyfish in locomotion. The preceding chapters have each looked at a different part of the question. I began with a detailed description of the kinematics of four jellyfish gaits. I described a resonant gait, in which the animals begin a new contraction just as die previous has reached its maximum reinflation, and showed that die jellyfish gain an amplitude boost from die resonant recoil. I showed tiiat the simple harmonic oscillator model suggested by DeMont & Gosline (1988c) could not explain the observed behaviour, concluding that a non-linear oscillator model was required. The mechanics behind this non-linear resonant gait provided die motivation for the rest of the thesis. Armed witii measurements made in Cliapter 3 of the material properties of the mesogleal matrix and the radially oriented reinforcing fibres, I developed a mathematical model of the jellyfish geometry to predict die mechanical behaviour of the locomotor system. In the last chapter, based on the assumptions tiiat the material properties of the mesoglea would not change with body size, and that jellyfish dimensions scaled geometrically with size, I extended die model, correctly predicting die mechanical behaviour of jellyfish of all sizes. Pabst (1996) discussed die importance of non-linear elasticity in aquatic locomotor systems. Aquatic animals whose muscles and springs are arranged in parallel need the non-linearity to ensure they aren't fighting witii the springs while trying to generate useful forward thrust. Gosline & Shadwick (1983a) and DeMont & Gosline (1988c) suggest tliat the non-linearity exists to allow the muscles to create uncompromised forward tiirust, posing little resistance to motion during die early stages of the stroke cycle, when tiirust is produced most efficiently, then storing energy in the springs at a time in the stroke cycle when die muscles are unable for hydrodynamic reasons to generate useful tiirust. This energy storage is hugely important for the jellyfish, which lacks any other means of refilling its bell to start the next contraction. In the squid, a second set of muscles exists to refill die mantie cavity, so it is not as constrained as die jellyfish, but as Gosline & Shadwick (1983a) point out, die addition of the spring makes it possible for die large circular swimming muscles to power part or most of the refilling stage. Indeed it would appear tiiat most of the refilling during normal swimming is powered by elastic energy (MacGillivray et al. 1999). The cost in mass of adding the collagen necessary to store die energy elastically is ininimal compared to die mass of die animal, and allows a reduction in the mass of the refill musculature. The reduction in overall body mass to produce die same thrust improves the animal's performance, and, inasmuch as improved performance can be related to improved fitness, ought to be favoured by natural selection. A second role for die non-linearity is in die control of the contraction amplitude. Both squid and jellyfish have to live inside their mantle/bell cavities, and too severe a contraction might eject part of the animal, or 108 at tlie very least damage vital tissues (Gosline & Shadwick 1983a). The cost to die animal in terms of sophisticated control systems would be great if it weren't for die non-linear spring system. Widiout die spring, the animal would have to actively monitor its diameter and turn on antagonistic muscles to stop the contraction before damage was done. Witii tlie non-linear spring, it can simply run tlie contraction until the muscle is no longer able to overcome tlie spring. Alternatively, it can begin a contraction, turn off the muscles once enough tiirust has been generated, and let momentum carry the stroke at the end until the spring force is enough to stop tlie movement. These are not new ideas. What is new, and worth thinking about, is the stiffness of tlie system at the turnaround and die resultant recoil frequency. Higher stiffness means higher natural frequency. This means that die system gets a high-frequency kick at die end of its jet stroke. Figure 6.1 shows tlie kinematics of recoil in jellyfish and squid. The recoil in both cases is powered by the energy stored in the spring (at least at the start - the squid may activate its radial muscles later in the stroke). The storage of energy late in the jet stroke may help pay for the reversal of angular momentum in die fluid and vorticity shedding observed by Ahlborn et al. (1991, 1997) to occur at the turnaround9. Tlie propellor (fin, fluke, margin, tunic) also has to be decelerated, then reaccelerated in the opposite direction - readily available, passive energy stores at die end of the jet stroke may facilitate tiiis process. Ahlborn et al. (1997) showed that by adjusting the timing of tlie bending of its tail at die end of a stroke, a fish might be able to harvest some tiirust from the vortices created by the motion of its body through the water. Intermediate bending stiffnesses in tlie tail were found to be the most efficient at tiiis process. It is possible that tlie non-linear spring of the jellyfish may be bringing the stiffness of the bell margin into this intermediate range, allowing die animal to gain an extra push off its own wake. This hypothesis is of especial interest in die light of recent work by Colin & Costello (2002) questioning jet-propulsion as a model for tlie locomotion of all hydrozoan jellyfish. They conclude that many jellyfish, particularly oblate ones, are in fact propelling diemselves using a drag-based mechanism, effectively rowing, or sculling through die water. Animals using tiiis mode of propulsion could derive an energetic benefit if their stiffness were tuned to take advantage of the extra push. All of the above indicates, as suggested by Pabst (1996), tiiat it is important for aquatic animals to have non-linear springs. But what is the origin of the non-linearity? The conclusion to draw from die work I've done in Chapter 4 is that it is tlie cylindrical geometry which is first and foremost responsible for the non-linearity of the system in deflation. Even with a perfectly linear matrix material, the elastic behaviour of a cylindrical object in deflation is non-linear. Actively swimming jellyfish have simply improved on die 9 Evidence tiiat jellyfish face tlie same fluid dynamics as die fish tail studied by Ahlborn et al. (1991) can be seen in Figure 7 of Colin & Costello (2002). 109 Jellyfish Squid 0.6 H 1 1 1 1 1 1 1 -4.0 4.4 4.8 5.2 5.6 0.0 Time (s) Figure 6.1. Pulse kinematics. A, Jellyfish. B, Squid, reproduced from Gosline & Shadwick (1983a). Note the non-sinusoidal shape of the pulse. The initial radial decrease is obviously due to the action of the swimming muscles, but in both cases, the beginning of the recovery phase is powered uniquely by the non-linear springs in die animals' locomotor apparatus. The slope is initially high, tiien drops off as the stiffness of the spring decreases as the system returns to its resting dimensions. This rapid turnaround at maximum extension may help to reverse and shed the bound vortex. - , , , h 0.6 0.4 0.8 1.2 1.6 Time (s) 110 design. The anisotropy introduced by the radially-oriented fibres increases the non-linearity. The increased stiffness at high strains makes it possible for die animal to further reduce die antagonism of the muscles during tiirust production through the stiffness-reducing action of tlie joints, but still store enough energy to power die refilling of its bell cavity. Tlie locomotor system of the squid is significantly more complex, with its three sets of fibrous elastic networks (Gosline & Shadwick 1983b), radial and circumferential muscle (Bone et al. 1981) and two muscle fibre types (Mommsen et al. 1981), but shares with the jellyfish its underlying cylindrical geometry. No equivalent measurements to diose reported here or by DeMont & Gosline (1988a) of the elastic behaviour of the whole structure have yet been made on squid mantie. Studies have been done on die mechanical behaviour of isolated samples of mantle tissue and shown it to be highly non-linear (Gosline & Shadwick 1983a, MacGillivray et al. 1999). Tlie combination of the cylindrical geometry and non-linear mantle material will produce an overall meclianical behaviour qualitatively similar to Figure 4.5. Stiffness will be low during die low strain part of the curve, tiien rise suddenly after some critical strain, likely near to die 12% reported by MacGillivray et al. (1999) as die strain at which die material begins to get stiff. Though joint-like structures are absent from the squid, the non-linearity of the mantle tissue provides an equivalent critical strain. Despite die similarity of their geometry, jellyfish and squid seem to function very differently. One is a resonator, die other is not. Tlie mass of fluid entrained by the refill stroke in squid and jellyfish of similar size and shape will be die same. However, die stiffness of die spring in die two cases is very different. Squid are some 3000x stiffer tiian jellyfish, even in the low-stiffness region of the non-linear curve. The highly compliant jellyfish spring turns out to be tuned such that die animal is able to take advantage of the energy savings associated with resonance. The squid, on the other hand, witii its much higher stiffness, due in part to its requirement for a more robust structure to withstand the high jet pressures first reported by Ward & Wainwright (1972), is unable to take advantage of die potential energy savings, as tlie resonant frequency of its locomotor system would be in the range of 15-30 Hz, far above the frequency range in which jet propulsion is efficient. DeMont & Gosline (1988c) first proposed that jellyfish might resonate. Tlie success of their model was however limited by its reliance on linear mathematics. Tlie inclusion of tlie non-linearity in Chapter 5 of this work made it possible to predict the kinematics of die mounted preparation. This would not have been possible with die linear model. Building on the success of the model at predicting tlie beliaviour of the mounted preparation, I adjusted (lie boundary conditions and was able to predict the kinematics of the tethered, but otherwise free-swimming animal reported by DeMont & Gosline (1988c). The addition of a realistic model of the added mass and some assumptions about the scaling of the animal with body length made it possible to predict die behaviour of jellyfish of all sizes. I l l The model additionally makes die prediction tiiat the resonant frequency should be a function of the driving amplitude. This provides the jellyfish with a simple control mechanism: swimming frequency can be controlled simply by the force of contraction. It also means tiiat die jellyfish can gain die benefit of resonance at multiple frequencies. Polyorchis does not normally adjust its force of contraction (Spencer 1995), however, other animals such asAglantha digitate, which make use of a high-frequency escape gait (3Hz Donaldson et al. 1980), may be able to take advantage of this phenomenon. Resonance has been proposed to explain die behaviour of odier aquatic organisms. The linear model seems to suffice for scallops (DeMont 1990, Cheng et al. 1996), since the spring in die hinge is itself linear. The linear resonance model, diough it came close, was insufficient to explain dolphin swimming (Pabst 1999). The effective stiffness of the elastic structures in the peduncle and tail of the dolphin is known to be non-linear (Long et al. 1997), so it is possible tliat die application of a model similar to die one proposed here for jellyfish might solve the problem. This diesis has been an integrated study of the locomotor system of die hydromedusan jellyfish Polyorchis penicillatus. The component parts of the jellyfish were assembled into a mathematical model which correctly predicted the dynamic behaviour of the system as a whole. Parts of the model were necessarily specific to the geometry of prolate hydromedusan jellyfish, and Polyorchis in particular, but much of the mathematics is immediately transferable to other systems, as I have suggested above for die squid and dolphin. Our collective understanding of the meclianics of aquatic locomotion has improved gready in recent years, particularly as numerical and analytical models acquire the sophistication to make integrative studies such as this one possible. My research will have been successful if it has laid the groundwork for future studies of the behaviour of highly non-linear elastic systems. 112 References AARON, B.B. AND GOSLINE, J.M. (1981). Elastin as a random-network elastomer, a mechanical and optical analysis of single elastin fibres. Biopolymers 20: 1247-1260. AGASSIZ, L. (1862). Contributions to die natural history of the United States of North America 4: 1-380. AHLBORN, B., CHAPMAN, S., STAFFORD, R., BLAKE , R.W. AND HARPER, D.G. (1997). Experimental simulation of the tiirust phases of fast-start swimming of fish. J. exp. Biol. 200: 2301-2312. AHLBORN, B., HARPER, D. G., BLAKE , R. W., AHLBORN, D. AND CA M , M. (1991). Fish without footprints. J. theor. Biol. 148: 521-534. ALEXANDER, R.McN. (1962). Visco-elastic properties of the body-wall of sea anemones. J. exp. Biol. 39: 373-386. ALEXANDER, R . M C N . (1974). Mechanics of jumping by a dog. J. Zool. (Lond.) 173: 549-573. ALEXANDER, R . M C N . (1976). Mechanics of bipedal locomotion, pp. 493-504 in: Spencer-Davies, P. (ed.). Perspectives in Experimental Biology, vol. 1. London: Academic Press. ALEXANDER, R . M C N . (1977). Swimming, pp. 222-248 in: Alexander, R.McN. and Goldspink, G. (eds.). Mechanics and Energetics of Animal Movement. London: Chapman and Hall. ALEXANDER, R . M C N . (1984). Elastic energy stores in running vertebrates. Am. Zool. 24: 85-94. ALEXANDER, R . M C N . (1987). Bending of cylindrical animals with helical fibers in tiieir skin or cuticle. J. theor. Biol. 124:97-110. ALEXANDER, R . M C N . AND VERNON, A. (1975). Mechanics of hopping by kangaroos (Macropodidae). J. Zool. (Lond.) 177: 265-303. ARAI , M. AND BRINCKMANN-VOSS, A. (1980). The hydromedusae of British Columbia and Puget Sound. Can. Bull. Fish, aquat. Sci. 204: 1-192. ARKETT, S.A. (1984). Diel vertical migration and feeding behavior of a demersal hydromedusan (Polyorchispenicillatus). Can. J. Fish, aquat. Sci. 41: 1837-1843. ARKETT, S.A. (1985). The shadow response of a hydromedusan (Polyorchis penicillatus): Behavioral mechanisms controlling diel and ontogenic vertical migration. Biol. Bull. 169: 297-312. BATCHELOR, G.K. (1967). An introduction to fluid mechanics. Cambridge: Cambridge University Press. BAUER, V . (1927). Die Schwimmbewegungen der Quallen und ihre reflektorische Regulierung. Z. vergl. Physiol. 5: 37-69. BLAKE , R.W. (1981). Mechanics of drag-based mechanisms of propulsion in aquatic vertebrates. Symp. Zool. Soc. Lond. 48: 29-52. BLICKHAN, R. AND CHENG, J.-Y. (1994). Energy storage by elastic mechanisms in the tail of large swimmers: A re-evaluation. J . theor. Biol. 168: 315-321. BONE, Q., PULSFORD, A. AND CHUBB, A.D. (1981). Squid mantie muscle. J. Mar. Biol. Assoc. U.K. 61: 327-342. BOUILLON, J. ANDCOPPOIS, G. (1977). Etude comparative de la mesoglee des cnidaires. Cah. Bio. Mar. 18: 339-368. BOUILLON, J. AND VANDERMEERSSCHE, G. (1957). Structure et nature de la mesoglee des hydro- et scyphomeduses. Annls. Soc. r. zool. Belg. 87: 9-25. BRINCKMANN-VOSS, A. (1977). The hydroid of Polyorchis penicillatus (Eschscholz) (Polyorchidae, Hydrozoa, Cnidaria). Can. J. Zool. 55: 93-96. BRINCKMANN-VOSS, A. (2000). The hydroid and medusa of Sarsia bella sp. nov. (Hydrozoa, Anthoatiiecatae, Corynidae), with a correction of the "life cycle" of Polyorchis penicillatus (Eschscholz). Sci. Mar. 64 (suppl. 1): 189-195. CAVAGNA, G.A. (1969). Travail mecanique dans la marche et la course. J. Physiol. (Paris) 61, suppl. 1: 3-42. CAVAGNA, G.A. (1975). Force platforms as ergometers. J. appl. Physiol. 39: 174-179. CAVAGNA, G.A. ANDMARGARIA, R. (1966). Mechanics of walking. J . appl. Physiol. 21: 271-278. CHAPMAN, G. (1953). Studies on the mesoglea of coelenterates. J. exp. Biol. 30: 440-451. CHAPMAN, G. (1958). The hydrostatic skeleton in the invertebrates. Biol. Rev. 33: 338-371. CHAPMAN, G. (1959). The mesoglea of Pelagia noctiluca. Q. J. Microsc. Sci. 100: 599-610. CHAPMAN, G. (1966). The structure and functions of the mesoglea. Symp. Zool. Soc. London 16: 147-168. 113 CHENG, J.-Y. AND BLICKHAN, R. (1994). Bending moment distribution along swimming fish. J. theor. Biol. 168: 337-348. CHENG, J.-Y., DAVISON, I.G. ANDDEMONT, M . E . (1996). Dynamics and energetics of scallop locomotion. J. exp. Biol. 199: 1931-1946. COLIN, S.P. AND COSTELLO, J.H. (2002). Morphology, swimming performance and propulsive mode of six co-occurring hydromedusae. J. exp. Biol. 205: 427-437. DANIEL, T.L. (1983). Mechanics and energedcs of medusan jet propulsion. Can. J. Zool. 61: 1406-1420. DANIEL, T.L. (1984). Unsteady aspects of aquadc locomotion. Am. Zool. 24: 121-134. DANIEL, T.L. (1995). Invertebrate swimming: integrating internal and external mechanics. In Symposia of the Society of Experimental Biology XL1X: Biological Fluid Dynamics (ed. CP. Ellington and T.J. Pedley), pp. 61-89. Cambridge: Company of Biologists Ltd. DEMONT , M.E. (1986). Mechanics of jet propulsion in a hydromedusean jellyfish. Doctoral dissertation, University of British Columbia, Vancouver. DEMONT , M E. (1990). Tuned oscillations in die swimming scallop Pecten maximus. Can. J. Zool. 68: 786-791. DEMONT , M.E. AND GOSLINE, J.M. (1988a). Mechanics of jet propulsion in the hydromedusan jellyfish, Polyorchis penicillatus. I. Mechanical properties of the locomotor structure. J. exp. Biol. 134: 313-332. DEMONT , M.E. AND GOSLINE, J.M. (1988b). Mechanics of jet propulsion in tlie hydromedusan jellyfish, Polyorchis penicillatus. II. Energetics of the jet cycle. J. exp. Biol. 134: 333-345. DEMONT , M.E. AND GOSLINE, J.M. (1988c). Mechanics of jet propulsion in the hydromedusan jellyfish, Polyorchis penicillatus. III. A natural resonating bell: The presence and importance of a resonant phenomenon in die locomotor structure. J. exp. Biol. 134: 347-361. DONALDSON, S., MACKIE, G .O. AND ROBERTS, A.O. (1980) Preliminary observations on escape swimming and giant neurones in Aglantha digitate (Hydromedusae: Trachylina). Can. J. Zool. 58: 549-552. DRUCKER, E.G. (1996). Tlie use of gait transition speed in comparative studies of fish locomotion. Am. Zool. 36: 555-566. DUFFING, G . (1918). Erzwungene Schwingungen bei verandlicher Eigenfrequenz. Vieweg Braunschweig. EMMETT, B., BURGER, L. AND CAROLSFELD, Y. (1995). An inventory and mapping of subtidal biophysical features of the Goose Islands, Hakai Recreation Area, British Columbia. Occasional Paper 3, BC Parks, Victoria. FRENCH, A.P. (1971). Vibrations and Waves. New York: W.W. Norton & Company, Inc. FUNG, Y.C. (1984). Structure and stress-strain relationship of soft tissues. Am. Zool. 24, 13-22. FUNG, Y.C. (1993). Biomechanics: mechanical properties of living tissues. New York: Springer Verlag. GLADFELTER, W.B. (1972). Structure and function of tlie locomotory system of Polyorchis montereyensis (Cnidaria, Hydrozoa). Helgoldnder wiss. Meeresunters. 23: 38-79. GLADFELTER, W.B. (1973). A comparative analysis of the locomotory systems of medusoid Cnidaria. Helgoldnder wiss. Meeresunters. 25: 228-272. GOSLINE, J.M. AND DEMONT , M.E. (1985). Jet-propelled swimming in squids. Scient. Am. 252: 96-103. GOSLINE, J.M. AND SHADWICK, R .E . (1983a). Tlie role of elastic energy storage mechanisms in swimming: an analysis of mantle elasticity in escape jetting in tlie squid, Loligo opalescens. Can. J. Zool. 61: 1421-1431. GOSLINE, J.M. AND SHADWICK, R .E . (1983b). Molluscan collagen and its mechanical organization in squid mantle, pp. 371-398 in: Hochaclika, P.W. (ed.) Biochemistry of mollusca. New York: Academic Press. GOSLINE, J.M., STEEVES, J.D., HARMAN, A.D. AND DEMONT , M.E. (1983). Patterns of circular and radial mantle muscle activity in respiration and jetting of the squid Loligo opalescens. J. exp. Biol. 104, 97-109. HOOKE, R. (1678). De Potentia Restitutiva. London. JOSEPHSON, R.K. (1999). Dissecting muscle power output. J. exp. Biol. 202: 3369-3375. KASAPI, M.A. AND GOSLINE, J.M. (1996). Strain-rate-dependent mechanical properties of the equine hoof wall. J. exp. Biol. 199: 1133-1146. LARSON, R .J. (1987). Costs of transport for the scyphomedusa Stomolophus meleagris L. Agassiz. Can. J. Zool. 65: 2690-2695. KING, M.G. AND SPENCER, A.N. (1981). Tlie involvement of nerves in die epidielial control of crumpling behaviour in a hydrozoan jellyfish. J. exp. Biol. 94: 203-218. 114 LIGHTHILL , SlRj. (1986). An informal introduction to theoretical fluid mechanics. Oxford: Oxford University Press. LILLIE , M.A., CHALMERS, G . W . A N D GOSLINE, J .M. (1994). The effects of heating on die mechanical properties of arterial elastin. Conn. Tiss. Res. 31: 23-35. LIN , Y.-C.J. A N D SPENCER , A.N. (2001). Localisation of intracellular calcium stores in the striated muscles of the jellyfish Polyorchis penicillatus: Possible involvement in excitation-contraction coupling. J. exp. Biol. 204: 3727-3736. L O N G , J.H.JR., PABST , D.A., SHEPHERD, W .R. A N D M C L E L L A N , W . A . (1997). Locomotor design of dolphin vetebral columns: bending meclianics and morphology of Delphinus delphis. J. exp. Biol. 200: 65-81. M A C G I L L I V R A Y , P.S., ANDERSON , E.J., WRIGHT, G . M . AND D E M O N T , M.E. (1999). Structure and mechanics of the squid mantie. J. exp. Biol. 202: 683-695. M A C K I E , G .O. A N D M A C K I E , G . V. (1967). Mesogleal ultrastructure and reversible opacity in a transparent siphonophore. Vie Milieu (A) 18: 47-71. M A C K I E , G .O. , A N D P A S S A N O , L.M. (1968). Epithelial conduction in hydromedusae. J. Gen. Physiol. 52, 600-621. M C C O N N E L L , C.J., D E M O N T , M.E. A N D WRIGHT, G . M . (1997). Microfibrils provide non-linear elastic behaviour in the abdominal artery of the lobster, Homarus americanus. J. Physiol. 499.2: 513-526. M C C O N N E L L , C.J., WRIGHT, G . M . AND D E M O N T , M.E. (1996). Tlie modulus of elasticity of lobster aorta microfibrils^  Experientia 52:918-921. M C H E N R Y , M.J., P E L L , C A . A N D L O N G , J.H.JR . (1995). Mechanical control of swimming speed: Stiffness and axial wave form in undulating fish models. J. exp. Biol. 198: 2293-2305. MEG ILL , W . M . (1996). Observations of Polyorchis penicillatus in Queen Charlotte Strait, BC, summer 1995. Vancouver: Coastal Ecosystems Research Foundation. Internet document: http://cerf.bc. ca/megill/pubs/polyorchisqcs. html MILLS , C E . (1981). Diversity of swimming behaviors in Hydromedusae as related to feeding and utilization of space. Mar. Biol. (Berl.) 64: 185-189. M O M M S E N , T.P., B A L L A N T Y N E , J., M A C D O N A L D , D., GOSLINE, J. A N D H O C H A C H K A , P W . (1981). Analogs of red and white muscle in squid mantle. Proc. Natl. Acad. Sci. 78: 3274- 3278. O ' D O R , R K. (1988). Tlie forces acting on swimming squid. J exp. Biol. 137: 421-442. PABST , D.A. (1999). Presentation to die 13th Biennial Conference on die Biology of Marine Mammals, Maui, HI, November 1999. PABST , D.A. (1996). Springs in swimming animals. Am. Zool. 36: 723-735 P A C K A R D , A. AND T R U E M A N , E.R. (1974). Muscular activity of the mantle of Sepia and Loligo (Cephalopoda) during respiratory movements and jetting, and its physiological interpretation. J. exp. Biol. 61, 411-419. PASSANO , L.M., MACK IE , G .O. AND P A V A N S DE C E C C A T T Y , M. (1967). Physiologie de comportement de l'hydromeduse Sarsia tubulosa Sars. Les systemes des activites spontanees. C.R. Acad. Sc. Paris D Zoologie : 614-617. PRESS, W . H . , T E U K O L S K Y , S.A., VETTERL ING, W .T . A N D F L A N N E R Y , B.P. (1992). Numerical recipes in C -The art of scientific computing. 2nd Edition. Cambridge: Cambridge University Press. R E B E R - M U L L E R , S., SPISSINGER, T., SCHUCHERT , P., SPRING, J. AND SCHMID , V. (1995). An extracellular matrix protein of jellyfish homologous to mammalian fibrillins forms different fibrils depending on die life stage of the animal. Dev. Biol. 168: 662-672. REES, J.T. AND L A R S O N , R.J. (1980). Morphological variation in tlie hydromedusa genus Polyorchis on the west coast of North America. Can. J. Zool. 58: 2089-2095. R O A C H , MR. A N D B U R T O N , A.C. (1957). The reason for the shape of the dispensability curves of arteries. CanJ. Biochem. Physiol. 35: 681-690. ROMANES, G.J. (1876). Preliminary observations on die locomotor system of medusae. Phil. Trans. R. Soc. Lond. (Biol.) 166: 269-313. SATTERLIE , R.A. A N D SPENCER , A.N. (1983). Neuronal control of locomotion in hydrozoan medusae. J. comp. Physiol. A 150: 195-206. SHADWICK , R.E. A N D GOSLINE, J .M. (1985). Mechanical properties of the octopus aorta. J. exp. Biol. 114: 259-284. SINGLA , C L . (1978a). Locomotion and neuromuscular system of Aglantha digitate. Cell Tiss. Res. 188: 317-327. 115 SINGLA, C L . (1978b). Fine structure of the neuromuscular system of Polyorchis penicillatus (Hydromedusae, Cnidaria). Cell Tiss. Res. 193: 163-174. SPEARING, S.M. (2001). M21 Generalized Hooke's Law. Boston: Massachusetts Inst, of Technology. Internet: http://web.mit.edu/16.unifwd/www/FALUmaterials/M21_M22_01_Notes.pdf. SPENCER, A.N. (1978). Neurobiology of Polyorchis. I. Function of effector systems. J. Neurobiol. 9: 153-157. SPENCER, A.N. (1979). Neurobiology of Polyorchis. II. Structure of effector systems. J. Neurobiol. 10: 95-117. SPENCER, A.N. (1982). The physiology of a coelenterate neuromuscular synapse. J. comp. Physiol. A 148: 353-363. SPENCER, A.N. (1995). Modulatory mechanisms at a primitive neuromuscular synapse: Membrane currents, transmitter release and modulation by transmitters in a cnidarian motor neuron. Am. Zool. 35: 520-528. SPENCER, A.N. AND ARKETT, S.A. (1984). Radial symmetry and the organization of central neurones in a hyrdozoan jellyfish. J. exp. Biol. 110: 69-90. SPENCER, A.N. AND SATTERLIE, R.A. (1981). The action potential and contraction in subumbrellar swimming muscle of Polyorchis penicillatus (Hydromedusae). J. Comp. Physiol. A 144: 401-407. SPENCER, A.N. AND SCHWAB, W.E. (1982). Hydrozoa. pp. 73-148 in: Shelton, G.A.B. (ed.) Electrical conduction and behaviour in "simple" invertebrates. New York: Clarendon Press. STELLE, L.L. (2001). Behavioral ecology of summer resident gray whales (Eschrichtius robustus) feeding on mysids in British Columbia, Canada. Doctoral dissertation, University of California at Los Angeles. STELLE, L.L. AND MEGILL, W.M. (in prep.). Penicillate jellyfish, Polyorchis penicillatus, predation on mysids, Holmsimysis sp., in Queen Charlotte Strait, BC. TAYLOR, CR. (1977). The energetics of terrestrial locomotion and body size in vertebrates, pp. 127-142 in: Pedley, T.J. (ed.) Scale effects in animal locomotion. London: Academic Press. THURMOND, F.A. AND TROTTER, J.A. (1996). Morphology and biomechanics of the microfibrillar network of sea cucumber dermis. J. exp. Biol. 199: 1817-1828. VALIDYNE ENGINEERING. (2002). Theory of operation - VR pressure transducers. Northridge, CA: Validyne Engineering, http://www.validyne.com/theory.htm. VOGEL, S. (1985). Flow-assisted shell reopening in swimming scallops. Biol. Bull. 169: 624-630. WARD , D.V. AND WAINWRIGHT, S.A. (1972). Locomotory aspects of squid mantle structure. J. Zool., Lond. 167: 437-449. WAINWRIGHT, S.A., BIGGS, W.D., CURREY, J.D. AND GOSLINE, J.M. (1976). Mechanical design in organisms. London: Edward Arnold. WARDLE, C.S., VIDELER, J.J. AND ALTRINGHAM, J.D. (1995). Tuning in to fish swimming waves: Body form, swimming mode and muscle function. J. exp. Biol. 198: 1629-1636. WEBB , P. W. (1971a). The swimming energetics of trout. I. Thrust and power output at cruising speeds. J. exp. Biol. 55: 521-540. WEBB , P.W. (1971b). The swimming energetics of trout. II. Oxygen consumption and swimming efficiency. J. exp. Biol. 55: 489-520 WEBB , P.W. (1978). Fast-start performance and body form in seven species of teleost fish. J. exp. Biol. 74: 211-216. WEBB , P W. (1983). Speed, acceleration and maneuverability of 2 teleost fishes. J. exp. Biol. 102: 115-122. WEBB , P.W. (1984). Body form, locomotion and foraging in aquatic vertebrates. Am. Zool. 24: 107-120. WEBB , P.W. AND GERSTNER, C L . (2000). Fish swimming behaviour: predictions from physical principles. pp. 59-77 in: Domenici, P. and Blake, R.W. (eds.) Biomechanics in Animal Behaviour. Oxford: BIOS Scientific Publishers Ltd. WEBER, C. AND SCHMID, V. (1985). The fibrous system in the extracellular matrix of hydromedusae. Tissue Cell IT. 811-822. WOLINSKY, H. AND GLAGOV, S. (1964). Structural basis for the static mechanical properties of the aortic media. Circulation Res. 14: 400-413. WRIGHT, D. (2001). Notes on design and analysis of machine elements. Perth: University of Western Australia, Dept. of Mechanical and Materials Engineering. Internet document: httpJ/www. mech. uwa. edu. au/DA Notes/cylinders/thick/thick.html. YOUNG, W.C. (1989). Roark's formulas for stress and strain, 6th Edition. Toronto: McGraw-Hill. 116 ZAR , J.H. (1984). Biostatistical analysis, 2nd ed. Englewood Cliffs, NJ: Prentice Hall. ZEEMAN, E.C. (2000). Duffing's equation: catastrophic jumps of amplitude and phase. Notes from a lecture given atUT San Antonio, March 31, 2000. Internet document: http://www. math. utsa. edu/ecz/duffmg. html. 


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