Stiffness and Damping Forces in the Intervertebral Joints of Blue Marlin (Makaira Nigricans)

Because the vertebrate body is mechanically complex, recent attempts to measure and model the transmission of muscular force and power in swimming fishes have remained incomplete (Hess and Videler, 1984; Johnsrude and Webb, 1985; Van Leewen et al. 1990). For example, while up to 94 % of muscle force may be lost in transmission to the tail (Johnsrude and Webb, 1985), the contribution of individual structures is still unknown (Van Leewen et al. 1990). One structure often implicated in force transmission -because of its connections to the lateral musculature, its axial position and its varied morphology - is the backbone (Home, 1809; Gregory and Conrad, 1937; Rockwell et al. 1938; Symmons, 1979; Wainwright, 1983).

Unfortunately, the forces transmitted by the backbone during swimming cannot be modeled from existing descriptions of the mechanical properties of fish backbones. This is because mechanical tests of the backbones of Norfolk spot Leiostomus xanthurus, skipjack tuna Katsuwonus pelamis (Hebrank, 1982) and blue marlin Makaira nigricans (Hebrank et al. 1990) were quasi-static, that is without specific reference to bending rate. Since the rate at which biological materials are deformed may drastically alter their mechanical properties (Wainwright et al. 1976), the mechanical function of the backbone over the range of tailbeat frequencies that occur during swimming can only be understood using mechanical properties that are measured dynamically. Therefore, the goals of this study are (1) to determine the force-transmitting capabilities of the backbone when tested at in vivo bending rates, (2) to determine whether the mechanical properties of the backbone vary with changes in bending amplitude, frequency and position along the axis, and (3) to describe how the mechanical properties are correlated with vertebral morphology.

This mechanical study focuses on the intervertebral joints, which, because of their relative flexibility compared to the vertebral centra, have been proposed as functionally important structures in the bending backbone (Home, 1809). The backbone I have chosen, because of its extremely robust structure (Cuvier and Valenciennes, 1831), is the vertebral column of the blue marlin Makaira nigricans. Each of the 23 intervertebral joints consists of intervertebral ligaments, which make up the biconic capsule connecting the centra (for teleosts in general see Symmons, 1979), as well as the large overlapping processes of the vertebrae (zygapophyses, neural and hemal spines) and their fibrous interconnections. Like all species of the family Istiophoridae, the neural and hemal spines have been modified into large, flat, thin plates that overlap the intervertebral joint and interlock via fibrous connective tissues with the expanded zygapophyses (Fig. 1).

This spectacular vertebral morphology has suggested a potential spring-like function to previous investigators (Gregory and Conrad, 1937; Rockwell et al. 1938; Hebrank et al. 1990). A spring transmits a force that is proportional to its change of shape or deformation. However, in addition to spring forces, even simple mechanical systems in motion also have damping forces, which retard motion, and inertial forces, which act in opposition to spring forces.

I tested nine of the 23 intervertebral joint positions of the blue marlin: precaudal joint positions 3, 5, 7 and 9, transitional joint position 11 (between the last precaudal and first caudal vertebrae) and caudal joint positions 13, 15, 17 and 19. The number of each intervertebral joint corresponds to the number of the vertebra anterior to it. Six joints of positions 3, 5, 7,13,17 and 19 and five joints of positions 9,11 and 15 were taken from six fresh blue marlin collected from local fish houses in Kailua-Kona, Hawaii. The marlin were caught on hook and line and put into cold storage no longer than 8h after capture.

The intact marlin weighed 29.3, 48.5, 79.2, 83.0, 93.4 and 96.6kg (mean weight 71.7 kg) and were stored for 24–72 h at 4°C. Since the heads were removed before I received the backbones, I used the overall weight of each marlin to estimate its length from the anterior tip of the lower jaw to the fork of the tail (E. C. Cyr, J. M. Dean, I. Jehangeer, M. Nallee, unpublished data). From lowest to highest weight, the estimated lengths were 1.40m, 1.65m, 1.90m, 1.95m, 2.00m and 2.03m (mean length 1.82 m). Backbones, once removed, were kept bathed in marine teleost saline (Pantin, 1964) and on ice until testing.

Intervertebral joints, left attached to adjacent vertebrae (as in Fig. 1), were tested individually. Each joint was warmed to room temperature (27–32°C) before, and continually bathed in saline during, the mechanical tests.

For each intervertebral joint, angular frequency, ω, and bending amplitude, θ₀, were varied and bending moment, M₀, and phase lag, δ, were measured in response. The bending motion was sinusoidal and varied over a range of bending frequencies (0.5–5 Hz) that included the range of tailbeat frequencies for hooked and free-swimming marlin (approximately 1–2 Hz) seen on underwater video tapes taken by Stephen Wainwright at Duke University and Sharkbait Productions, Kona, Hawaii. The following discussion uses both bending frequency,f, in Hertz, and angular frequency, ω, in radians per second. Since the curvature of marlin during swimming could not be quantified from the videotapes, the range of bending amplitudes for each intervertebral joint was roughly estimated to be between 0 and 5.0°. For mechanical testing, two bending amplitudes, ±3.3° and ±5.0°, were used.

The bending machine in Fig. 2 converted the rotary motion of a motor into a reciprocating bending motion with an amplitude that varies as a sinusoidal function of time. In order to be tested, each intervertebral joint was positioned with its lateral bending axis, presumed to run through the middle of the joint, collinear with the axis of the shaft D, perpendicular to the page in Fig. 2.

Bending frequencies ranged in 0.5 Hz increments from 0.5 Hz to 5 Hz. Occasional zero-offset errors caused the maximum bending frequency for some tests to reach 7 Hz. The resonance frequency of the machine, with and without loaded intervertebral joints, was approximately 12 Hz. Bending frequency was controlled with a Minarik model SL-52 speed control attached to the motor and was measured directly using a Schaevitz Engineering model R30D rotary variable differential transducer (RVDT) (125 mV degree⁻¹, 0.25% linearity) mounted coaxially with the rotating shaft (point D, Fig. 2). The displacement signal from the RVDT was digitally sampled at 500 points per second using a National Instruments model NB-MIO-16 16 bit analog-to-digital converter and Lab VIEW software with an Apple Macintosh Ilex. For any chosen bending frequency, fluctuations in motor speed and measurement imprecision combined for a repeatability, or imprecision, in angular frequency of ±2% (range) about the mean with a resolution, or inaccuracy, of 0.002rads⁻¹.

Bending amplitude was controlled by changing the length of the input linkage (line ⁠, Fig. 2). Resolution of the amplitude measurements was 0.006 rad.

The bending moment causing the sinusoidal motion (M₀, Fig. 3) was measured with two 120 Ω foil strain gauges mounted on the steel output linkage (line ⁠, Fig. 2). The strain gauges were calibrated using calibrated weights. The moment arm of the output linkage was 11.7 cm and at that distance the gauges were sensitive to a deformation caused by a force of 0.1 N, which yields a moment resolution of approximately 0.01 N m. The strain gauge signal was generated using a bridge amplifier designed by Steven Vogel of Duke University with an Intersil commutating auto-zero instrumentation amplifier, model number CJN7605 (rated to 10Hz). The bending moment signal was sampled at 500 points per second in the same manner as the angular deformation with a 3 % range of repeatability.

The other measured response to dynamic bending deformation was the phase lag (δ, Fig. 3), the time between the maximum angular displacement and the maximum bending moment. It was measured as the time between the zerocrossings of the moment and displacement signals and was converted to radians to normalize for each bending frequency. Resolution of phase lag was 0.002 s, with a 10 % range of repeatability.

At each angular frequency for a given bending amplitude, angular stiffness, k, was computed by substituting the calculated value for I back into equation 3. This yielded a resolution of 0.01 Nm rad⁻¹ and a 5 % range of repeatability.

Each intervertebral joint was tested using the following experimental protocol. After being clamped into the bending machine, each joint was preconditioned for approximately 30 s at a bending frequency of 2 Hz. Preconditioning stabilizes the mechanical behavior of the joint, a process that yields repeatable dynamic properties in articular cartilage (Woo et al. 1987) and a stable moment amplitude (Fig. 3) after several bending cycles at each new test frequency in the joint. After the initial preconditioning, the joint was bent at each of the 10 bending frequencies, starting with 0.5Hz and increasing in 0.5Hz increments to 5.0Hz. This frequency sweep was made first at a bending amplitude of ±3.3° and then at a bending amplitude of ±5.0°. At each frequency, data were collected after at least 10 bending cycles.

For each frequency and amplitude, a computer file was created to store the simultaneous bending displacement and bending moment signals. I measured M₀, the amplitude of the bending moment curve, and δ, the time in seconds between maximum bending displacement and maximum bending moment (Fig. 3), using National Instrument’s LabVIEW software. Each value of M₀ and δ for each frequency and amplitude was the average of five successive measurements along the time axis. For any joint, the mechanical behavior was stable over time - data taken at a specific frequency during the testing protocol did not differ significantly from data taken afterwards.

This statistical analysis was designed to test the hypothesis that the mechanical properties, k and c, change as a result of changes in bending frequency, bending amplitude and joint position. In addition, variation due to size differences between marlin was taken into account.

Since k and c were measured repeatedly at successively higher bending frequencies on the same joint, the measurements were dependent through time. Thus, I could not compare regressions of k or c onto bending frequency using analysis of covariance (Sokal and Rohlf, 1981). Instead, I compared the means of the regression parameters for each joint position and bending amplitude using t- tests (Sokal and Rohlf, 1981) and the Bonferroni procedure to account for multiple comparisons (Wilkinson, 1989). This method is analogous to a profile analysis of variance (Simms and Burdick, 1988).

To account for between-animal variation in size, I repeated the analysis with length-specific regression parameters. Within a given marlin, the regression parameters were normalized by the length of the vertebral column from vertebra 2 to vertebra 20. For marlin with body weights of 29.3, 48.5, 79.2, 83.0, 93.4 and 96.6kg, these lengths, L, were, respectively, 88.0, 99.4, 116.7, 119.5, 122.8 and 108.6 cm.

The mechanical experiments at each joint position were replicated using six marlin. There were six replicates for joint positions 3, 5, 7, 13, 17 and 19 and five replicates for joint positions 9, 11 and 15.

Using r² values as the criterion, the best-fitting regression of angular stiffness, k, onto bending frequency was a linear regression. The same was true for damping coefficient, c. The regressions and all other statistics were calculated using the Systat Statistical Package, version 5.0 on the Apple Macintosh Ilex (Wilkinson, 1989).

For angular stiffness, k, the intercept of the y-axis can be thought of as the stiffness of the intervertebral joint at rest, the so-called static angular stiffness. The slope of the line is the frequency response of stiffness, the change in k with the rate of deformation. The intercept of damping coefficient, c, has no simple mechanical interpretation. However, the slope of the line is the frequency response of the damping coefficient.

I measured the following intervertebral and vertebral characters (Fig. 1): length of vertebral centra anterior and posterior to the joint, length of intervertebral ligaments (Fig. 1A); transverse height and width of the intervertebral capsule; and transverse lateral thickness of the intervertebral capsule (Fig. 1B). All morphometric measurements were made using Monostat model 6921 calipers with an accuracy of ±0.1 mm. Imprecision of the morphometries ranges by ±6% about the mean.

The lengths of the vertebral centra, anterior and posterior to the joint, were chosen because (1) each centrum can be thought of as a lever arm that bends the intervertebral joint and (2) each centrum has zygapophyses and neural and hemal spines that affect stiffness (Hebrank et al. 1990). The length of the intervertebral ligaments was chosen because it is a direct measure of the axial size of the joint, which should scale with stiffness. Transverse height, width and lateral thickness of the intervertebral capsule were chosen because similar dimensions are used to calculate the second moment of area, which determines flexural stiffness in manmade structures (Den Hartog, 1949).

To determine which subset of these morphological characters was the best predictor of k and c, I performed step-wise multiple regressions. The intercepts and slopes of the regression of angular stiffness, k, and damping coefficient, c, onto bending frequency were used as separate dependent variables and the morphological characters were the independent variables. Independent variables that were not normally distributed were log-transformed.

The means and standard deviations of the intercepts and slopes of the regressions of angular stiffness, k, and damping coefficient, c, onto bending frequency, f, are given for each intervertebral joint at each bending amplitude in Table 1. The following significant differences were found (α<0.05 using the Bonferroni procedure).

The means of angular stiffness intercepts at bending amplitude ±3.3° are less than the means at amplitude ±5.0° for joint positions 11, 13, 17 and 19. The same trend holds for the length-specific intercepts with the addition of joint position 15.

At a bending amplitude of ±3.3°, the mean of the intercept of joint position 13 is greater than that of joint position 7. The mean of joint position 17 is greater than the means of joint positions 3, 7 and 13. For length-specific intercepts at this bending amplitude, the mean of joint position 13 is greater than the means of positions 3 and 7, the mean of joint position 15 is greater than the means of positions 3 and 5, the mean of joint position 17 is greater than the means of positions 3,5,7 and 13, and the mean of joint position 19 is greater than the means of positions 3 and 5.

At a bending amplitude of ±5.0°, the mean of the intercept of joint position 13 is significantly greater than the mean of joint position 3. At the same amplitude, the mean of the intercept of joint position 17 is greater than that of joint positions 3 and 7. For length-specific intercepts at this amplitude, the mean of joint position 13 is greater than the mean of position 3, the mean of joint position 17 is greater than the means of positions 3, 5 and 7, and the mean of joint position 19 is greater than the mean of joint position 3.

The mean of angular stiffness slopes at a bending amplitude of ±3.3° is significantly greater than the mean of angular stiffness slopes at an amplitude of ±5.0° for joint positions 13 and 19. The means of the length-specific slopes follow the same pattern.

The means of the slopes of joint positions 7–19 at a bending amplitude of ±5.0° are significantly less than zero.

At a bending amplitude of ±5.0°, the mean of the damping coefficient intercept of joint position 19 is significantly greater than the mean of the intercept of joint position 7. For length-specific intercepts, there are no significant differences.

At a bending amplitude of ±5.0°, the mean of the damping coefficient slope of joint position 19 is significantly greater than the mean of the slope of joint position 3. For length-specific damping slope, the mean of joint position 19 is greater than the mean of joint position 3 at both bending amplitudes.

At both bending amplitudes, the means of the slopes of the damping coefficients for joint positions 3 and 5 are significantly less than zero.

Angular stiffness and damping coefficient as a function of bending frequency are shown by bending amplitude and joint position in Figs 4, 5 and 6.

The morphometrics of the intervertebral joints and associated vertebrae are given in Table 2. The lengths of the centra anterior and posterior to the intervertebral joints increase until joint 15–17, and then decrease in length towards the tail. The transverse lateral thickness of the intervertebral joint is the only other measurement to show a similar pattern. The transverse height of the joint decreases from head to tail. The axial joint length and the transverse width of the joint vary only slightly.

The morphological characters that best predict the intercepts and slopes of the regression of angular stiffness and damping coefficient onto bending frequency are given in Table 3. At both bending amplitudes, 88% of the variation in the intercept of angular stiffness is explained by the variation in three morphological characters. For the slope of angular stiffness, only 13 % and 36 % of the variation, for amplitudes of ±3.3° and ±5.0°, respectively, is explained by the variation in two morphological characters. For damping coefficient, only the intercept at a bending amplitude of ±5.0° has more than 50% of its variation explained by variation in the morphological characters.

Examination of the standardized coefficients, which account for the differences in size between the morphological characters, shows that, for angular stiffness, the most important variable is the logarithm of the length of the anterior centrum. For the intercept of damping coefficient, length of the posterior centrum is most important. For the slope of the damping coefficient, the most important variables are width of the intervertebral capsule at ±3.3° and transverse thickness of the intervertebral ligaments at ±5.0°.

The equation of motion, which describes the relationship between force and motion, has been used to determine the mechanical properties that are important to the dynamic functions of intervertebral joints. This approach has yielded several insights. First, the mechanical properties of angular stiffness and damping coefficient vary with the motion of the joint. Second, there is regional variation in angular stiffness and in damping coefficient; this variation is correlated with structural variation. Third, the mechanics of the intervertebral joints are dominated by stiffness moments.

Bending motion changes the mechanical properties of intervertebral joints The angular stiffness of the intervertebral joints is a variable mechanical property, changing as a function of bending amplitude and frequency (Figs 4 and 5). In a swimming fish, bending amplitude increases along the body and tailbeat (bending) frequency changes with swimming speed (Videler and Hess, 1984; Webb, 1986). What effect might these changes in swimming motion have on the mechanics and functions of the intervertebral joints? With increasing bending amplitude, increases in angular stiffness of the intervertebral joints would, for each increment of angular displacement, cause the intervertebral joints to transmit proportionally more bending moment. This non-linear increase in stiffness with strain is called ‘spring hardening’ (Timoshenko et al. 1974) and is typical of statically deformed teleost intervertebral joints (Hebrank, 1982; Hebrank et al. 1990) and other connective tissues (Wainwright et al. 1976). In contrast, with increases in tailbeat frequency, the transmitted moment would actually decrease slightly, since the rate of increase with bending frequency of many of the intervertebral joint positions is negative.

This negative response of angular stiffness to increases in bending frequency must be viewed with some caution, since other viscoelastic polymers do not exhibit this behavior (Ferry, 1961). There are two classes of explanation for this phenomenon: (1) the intervertebral joints are built of novel materials or structures, or (2) the negative slope, seen only at bending amplitudes of ±5.0°, is a result of the 7 % drop in sensitivity of the bending machine at high frequencies (see Materials and methods). I favor the former explanation, since the smallest significant decrease from 0.5 to 5.0 Hz for a joint position is 13 %, while the largest is 21%.

In addition to increasing the transmitted moment, amplitude-dependent increases in the angular stiffness of the intervertebral joints may increase the speed of the undulatory waves propagated down the vertebral column. This hypothesis is based on the fact that the speed at which a mechanical vibration is propagated through a homogeneous solid is proportional to the square root of the material’s elastic modulus (Timoshenko et al. 1974). The elastic modulus is the proportionality constant of stress and strain, making it mechanically analogous to angular stiffness, k, which is the proportionality constant of bending moment and angular displacement (see equation 1).

In contrast to angular stiffness, damping coefficient, c, of the intervertebral joints is independent of the bending amplitude. However, damping coefficient does decrease with bending frequency at joint positions 3 and 5 (Table 1; Fig. 6). This frequency-dependent decrease in c is similar to the behavior of the mechanically analogous dynamic viscosity, η ′, as measured in amorphous viscoelastic polymers (Ferry, 1961). This drop in damping coefficient with increased bending frequency has two effects. First, the tangent of δ (see equation 2), which is an indicator of the proportion of energy dissipated (Wainwright et al. 1976), will not increase as quickly with frequency as it would if c were frequencyindependent. Nonetheless, the behavior of tanδ would be similar to that of viscoelastic polymers (Ferry, 1961; Wainwright et al. 1976). Second, since damping forces are proportional to the angular velocity, less damping force will be generated by the joint with increased frequency than would be generated if c were frequency-independent.

Angular stiffness, k, as represented by the intercept of its regression onto bending frequency, is greater in the caudal intervertebral joint positions (13–19) than in the precaudal joint positions (3–7). This anterior-posterior difference is particularly clear when the regression intercepts are normalized for differences in size between the marlin tested. In contrast, the intercept of damping coefficient differs only between joint positions 7 and 19, a distinction that disappears when the intercepts are normalized for size. As suggested above, angular stiffness may function in the regulation of the speed at which an undulatory wave propagates through an intervertebral joint. Considering the vertebral column as a whole, the increase of angular stiffness along the vertebral column may accelerate the undulatory wave as it travels along the body. It has been suggested that an axial gradient of mechanical properties is a necessary condition of aquatic undulatory locomotion (Blight, 1977). For example, Webb (1973) described a reversible stiffness gradient in the notochord of Branchiostoma that is correlated with the alternating backward and forward propagation of undulatory waves. In the context of this study, however, it should be remembered that, although the flexibility of the body is affected by the stiffness of the vertebral column, muscle and skin play an important role as well.

A flexible anterior vertebral column may also function in feeding. Underwater videotapes, taken by Sharkbait Productions (Kona, Hawaii) of a free-swimming blue marlin approaching a live bait, show the marlin swinging its head in an attempt to strike the bait with its bill. Other footage shows a hooked blue marlin whipping its head from side to side in an attempt to dislodge the hook. These observations suggest that the marlin uses its relatively flexible ‘neck’ to bludgeon prey with its bill or dislodge over-size prey.

The regional variations in angular stiffness and damping coefficient are correlated with the variation in a number of the vertebral characters (Fig. 1; Tables 2 and 3). The variation in angular stiffness is the most tightly correlated with vertebral characters, the most important of which, according to the standardized regression coefficients, is the logarithm of the length of the centrum anterior to the intervertebral joint. At the same time, the length of the centrum posterior to the joint is the most important correlate of the intercept of the regression of damping coefficient onto bending frequency.

Why is the length of the vertebral centra correlated with angular stiffness and damping coefficient? Since the bony centrum is not part of the joint, there must be unmeasured structures correlated with centrum length, or size in general, that are mechanical determinants of k and c. Likely candidates are the interlocking zygapophyses and neural and hemal spines (Fig. 1), all of which span the intervertebral joint and vary in size along the vertebral axis. The zygapophyses have a measurable effect on lateral stiffness, since removing them partially reduces quasi-static bending stiffness (Hebrank et al. 1990). It is interesting to note that such robust, interlocking processes are missing from the marlin’s sister taxa, the swordfish Xiphius gladius (Gregory and Conrad, 1937) and the tunas and mackerels (Fierstine and Walters, 1968). Based on this inter-familial variation, I would predict correlated differences in the angular stiffness and damping coefficient in these related taxa.

For all intervertebral joint positions and bending frequencies, the maximal stiffness moments during bending are always greater, within a frequency range of 0.5–5.0 Hz, than the maximal damping moments (Fig. 7). Values of the moments for in vivo strain rates (1–2 Hz) are intermediate with respect to the extreme frequencies shown. The maximal stiffness moment, M_k, for each joint position was computed as the product of the angular stiffness for that frequency (from Table 1) and angular displacement, when t is chosen so that angular displacement is maximal:

In addition to transmitting moments through the joint, the stiffness and damping moments cause the external moment creating the motion - be it of muscular or hydrodynamic origin - to do mechanical work. For a complete bending cycle, the work done by the external moment as a result of the stiffness and damping moments can be measured as the area under a power-time curve (Fig. 8). Power is calculated as the product of a force and a velocity, or, as in this angular case, the product of a moment and an angular velocity. The external power needed to overcome the stiffness moment at any time, t, is:

Because the intervertebral joints are dominated by stiffness moments, the work-related functions of the joint are to provide (1) an elastic energy storage mechanism during the negative work phase when the joint moves from a bent to an unbent position and (2) a muscle antagonist during the positive work phase when the joint moves from an unbent to a bent position. While there are many examples of the usefulness of energy return (Alexander, 1988), what are the functional consequences of energy loss in the intervertebral joints? One possible function of energy dissipation by the intervertebral joints may be dynamic stability. Meyhöfer and Daniel (1990) have argued that energy dissipation in muscle cells prevents high and potentially destructive internal strains in the tails of shrimp during escape locomotion. I suggest that the intervertebral joints, which develop substantial damping moments only at high tailbeat frequencies when locomotor power is maximal, may perform a similar function at high speeds in the blue marlin.

While moments external and internal to the intervertebral joints are the foci of this discussion, it is important to realize that, with one theoretical exception (Hess and Videler, 1984), the bending moments generated by muscles during locomotion remain enigmatic. In practice, muscular moments are difficult to measure because it is unclear what moment arms are used by the muscles. As this study demonstrates, sizable muscular moments could be transmitted by the vertebral column to the tail. In spite of increasingly sophisticated knowledge of the force, power and shortening velocities of muscle in swimming fish (e.g. Rome and Sosnicki, 1990; Rome et al. 1990), we still do not understand how the mechanical interactions of the muscle and vertebral column cause the fish to bend (Wainwright, 1983; Long et al. 1990).

 
• c

  damping coefficient (in kgm² rad⁻² s⁻¹)

 
• f

  bending frequency (in Hz)

 
• 1

  moment of inertia (in kgm² rad⁻³)

 
• k

  angular stiffness (in N m rad⁻¹)

 
• L

  length of vertebral column from vertebra 2 to vertebra 20 (in m)

 
• L_A

  length of the vertebral centrum anterior to the joint (in cm)

 
• L_J

  length of the intervertebral ligaments (in cm)

 
• L_P

  length of the vertebral centrum posterior to the joint (in cm)

 
• M_c

  maximal damping moment during a bending cycle (in N m)

 
• M_k

  maximal stiffness moment during a bending cycle (in N m)

 
• M₀

  amplitude of bending moment (in N m)

 
• P_c

  power of external moment needed to overcome damping moments (in W)

 
• P_k

  power of external moment needed to overcome stiffness moments (in W)

 
• t

  time (in s)

 
• T_H

  transverse height of the intervertebral capsule (in cm)

 
• T_w

  transverse width of the intervertebral capsule (in cm)

 
• T_T

  transverse lateral thickness of the intervertebral capsule (in cm)

 
• δ

  phase lag between bending displacement and bending moment (in s or rad)

 
• θ₀

  amplitude of bending displacement (in rad)

 
• ω

  angular frequency (in rads⁻¹)

I am indebted to Stephen A. Wainwright for challenging and inspiring discussions about backbones, gentle assistance in the field and laboratory, and constructive readings of the manuscript. I thank Steve Vogel for advice and guidance building the prototype of the dynamic bending machine. I thank John Hebrank for advice on refining later versions of the bending machine. I am grateful to Barbara Block for convincing me that marlin were worthy of study. For statistical advice, I thank Don Burdick and Mark Westneat. The field work in Hawaii was made possible by David Grobecker, Director of Pacific Ocean Research Foundation, and a grant from the Lerner-Gray Fund of the American Museum of Natural History. Manuscript readers who were of great help in early drafts and take no blame for the problems of the final version are Hugh Crenshaw, Mary and Jack Hebrank, Anne Moore, Margaret Ronsheim, V. Louise Roth, Kathleen Smith and Mark Westneat. Later versions were helped by two anonymous reviewers. Illustrations are by Rosemary Calvert. I was supported by a Morphology Fellowship from the Cocos Foundation of Indianapolis, Indiana.