Locomotor Design Of Dolphin Vertebral Columns: Bending Mechanics And Morphology Of Delphinus Delphis

Dolphins swim by rhythmically bending a variably flexible beam – their vertebral column. With the evolution of fully aquatic swimming behavior, the vertebral column of cetaceans has undergone comprehensive changes in function and structure compared with that of the group’s hypothesized terrestrial ancestors (Slijper, 1936). Recently, putative evolutionary intermediates have been unearthed, yielding, compared with terrestrial mammals, vertebral columns with reduced thoracic spinous processes, unfused and lengthened sacral vertebrae and robust lumbar neural spines (Thewissen et al. 1994; Gingerich et al. 1994). Thewissen et al. (1994) used these vertebral features to argue that early cetaceans swam in a manner similar to all extant representatives of the order – namely by dorsoventral bending of the axial skeleton. Yet we know little about the relationship of vertebral morphology – either skeletal or ligamentous – to the biomechanics of bending vertebral columns. The goal of the present study was to elucidate that relationship by examining the static and dynamic mechanical behaviors of intervertebral joint segments of the saddleback dolphin Delphinus delphis.

The vertebral column of cetaceans, as in all vertebrates, transmits forces that contribute to movement and that control the range and pattern of body deformations (e.g. Alexander et al. 1985; Bennett et al. 1987; Gal, 1992, 1993a; Hebrank, 1982; Hebrank et al. 1990; Long, 1992, 1995; Pabst, 1993; Slijper, 1936, 1946; Wainwright, 1983). These functions are controlled, in part, by the morphologies and mechanical properties of the segments that form the vertebral column: the vertebrae and the ligaments that serially connect them. For example, Hebrank et al. (1990) demonstrated that the neural and hemal arches, zygopophyses and intervening ligaments stabilize the intervertebral joints of blue marlin (Pisces: Makaira nigricans) against axial compression and lateral shearing – functions critical to the mechanical integrity of the vertebral column. In mammalian vertebral columns, Gal (1993b) demonstrated that the relationship between vertebral structure and bending mechanics is species-specific. For example, resistance to ventral flexion is controlled primarily by the ligamenta flava in some monkeys and wallabies and by intervertebral discs in tigers and jaguars (Gal, 1993b).

Vertebral morphologies and bending mechanics also vary regionally and may be important in controlling the pattern of force transmission and deformation along the body axis. Precaudal intervertebral discs of blue marlin are less stiff than the caudal joints – the increased stiffness may accelerate the undulatory wave during swimming (Long, 1992). The lumbosacral joint in a number of terrestrial mammals is significantly less resistant to bending than the lumbar–lumbar joints, a regional specialization that may enhance pelvic limb movement (Gal, 1993a).

The extent of bending varies both along the length of the animal and in the up-and downstrokes of many cetaceans, as shown by kinematic analyses of swimming motions and qualitative bending tests (e.g. Videler and Kammermans, 1985; Wainwright et al. 1987). Bottlenose dolphins (Tursiops truncatus) and a number of other toothed cetaceans swim by oscillating the caudal third of their body (Fish, 1993). The small deformations cranial to the dorsal fin make body bending difficult to measure except at the atlanto-occiptial joint (Pabst, 1993). As determined by manually bending a carcass, harbor porpoises (Phocoena phocoena) appear to have two ‘centers of rotation’, one at the lumbo-caudal joint and the other at the caudal joint at the insertion of the fluke blades (Parry, 1949). Slijper (1946) suggested that the perceived increased resistance to bending in the lumbar region was due to the relatively large second moment of area of the vertebral bodies (a cross-sectional shape factor used in engineering beam theory). Regional variations in bending and dorsoventral asymmetries have also been attributed to the increased length of spinous processes, extra articulating processes (Slijper, 1946) and the ventral longitudinal ligament (Parry, 1949).

No quantitative analysis exists that tests the relationship between any morphological feature and the bending mechanics of the vertebral column in any cetacean species (although see Bennett et al. 1987 for the mechanical properties of the caudal vertebral column in the harbor porpoise). The goals of the present study are (1) to characterize the static and dynamic mechanical properties of the intervertebral joint segments in the saddleback dolphin, (2) to determine whether regional variation and/or dorsoventral asymmetries in mechanical performance exist, (3) to describe how the mechanical properties are correlated with vertebral morphology, and (4) to hypothesize functional roles of the vertebral column in a swimming dolphin.

We investigated the vertebral columns of five saddleback dolphins, Delphinus delphis Linnaeus, at the United States National Museum (Table 1). These individuals were captured incidental to fishing operations in the north-west Atlantic Ocean, frozen immediately, and kept in that disposition until our dissections.

For bending tests, we removed single intervertebral joint segments from the backbone, each consisting of a cranial vertebra, an intact ligamentous joint and a caudal vertebra (Figs 1, 2A). These joint segments were bathed continuously in physiological saline (Pantin, 1964) as they thawed, warmed to room temperature and then underwent bending experiments. Eight joint segments were sampled from three regions of the backbone (Fig. 1): thoracic 4/5, thoraco-lumbar, lumbar 10/11, lumbo-caudal, caudal 7/8, caudal 13/14, caudal 20/21 and prefluke/fluke (where the two numbers, when used, refer to the axial positions of the cranial and caudal vertebrae, respectively). Prior to testing, joints were preconditioned for 5–10 bending cycles, after which the stiffness of the joints was repeatable within 10 % of the mean (see ‘Bending tests’ below).

To measure their mechanical properties, intervertebral segments were bent in a machine (Fig. 3), modified from Long (1992), which applied an angular strain (θ, rad) and measured the resulting bending moment (M, N m). To grip the segment, needle-nose locking pliers (Sears Craftsman) were secured to each of the four transverse processes of the two vertebrae close to the vertebral bodies; the bending moment was applied to the joint via these bony processes (see Fig. 2). The pliers, in turn, were mounted onto the machine so that the intervertebral segment was bent in the dorsoventral plane, with the axis of bending passing through the dorsoventral and rostro–caudal midpoint of the intervertebral disc (Fig. 3). The moment arm on the stationary side of the machine was kept constant for all joints and tests. The arrangement was visually inspected during preconditioning to ensure that there was no slippage between the pliers and the transverse processes.

For quasi-static bending tests, the bending moment was applied manually to the intervertebral joint via the input linkage (Fig. 3). Joints were bent in dorsal extension and ventral flexion. The bending moment was measured using a half-bridge foil strain gauge (120 Ω) configuration with each gauge glued onto either side of an aluminum bar. The gauges were excited by a bridge amplifier (Omega Engineering, model DMD-520). The angular displacement, θ, caused by the bending moment was monitored by a rotary-variable differential transducer (Schaevitz, model R30D). Voltage outputs from both transducers were digitally sampled (100 Hz) using an analog-to-digital converter (National Instruments, model NB-MIO-16L) and a microcomputer (Apple Macintosh, model IIcx).

For dynamic bending tests, the input linkage of the machine was attached to a motor in such a way as to produce a reciprocating, sinusoidal motion (Fig. 3). The bending frequency (Hz) was determined by selecting the rotational speed of the shunt-wound d.c. motor using an electronic controller (Minark Electric, model SL52). The maximal bending amplitude, θ₀, was controlled by altering the length of the input linkage (the distance between pivots A and D in Fig. 3).

Quasi-static bending tests were performed to measure the stiffness of the initial, or low-strain, response of the intervertebral joint. Initial stiffness, the slope of the loading curve (equation 1), provides a simple diagnostic of the bending resistance offered by the intervertebral segments as a function of the following independent variables: (1) region of the backbone (thoracic, lumbar, caudal, fluke), (2) direction of bending (extension, flexion) and (3) integrity of intervertebral ligaments (see Fig. 4A,B). The starting position of the joint chosen as zero displacement and moment was the position assumed by the mounted joint following a single displacement of the free end (pivot D, Fig. 3) of the input linkage; the resulting resonance of the joint produced angular oscillations about a reproducible position that we defined as the ‘straight’ or 0 ° angular postion. Reproducibility of this position was possible since the joints lacked neutral zones. It is unknown how this in vitro straight posture relates to the posture of the spine in vivo.

Each intervertebral segment was tested in the following order: intact (quasi-static), intact dynamic, interspinous ligaments cut (quasi-static) and intertransverse ligaments cut (quasi-static). The total elapsed time, from the first to the last test, ranged from 13 to 17 min. Note that owing to operational errors during some tests, not all joints are represented throughout the testing sequence (see Table 1); this uneven sampling is responsible, in some cases, for small apparent increases (less than 13 %) in initial stiffness following the cutting of ligaments (see Results). To account for possible changes in the material properties over the course of the experiment and for investigator error, we performed a repeatability analysis consisting of three separate quasi-static tests of an intact joint tested over 15 min (the average duration of the battery of tests); the joints were bathed with saline throughout and analyzed by three separate investigators. For time-dependent effects on material properties, no consistent directional changes in stiffness were detected between trials, and values varied by ±5 % (1 S.D., N=3) about the mean stiffness. To determine the magnitude of investigator error, three separate investigators independently analyzed the same data; the stiffness values from the three varied by ±6 % (1 S.D., N=3) about the mean. When the between-trials and between-investigator effects were combined, the stiffness values varied by ±10 % (1 S.D., N=9) about the mean.

For two individual dolphins (571449 and NY 08/10/91; see Table 1), we video-taped the joints as they underwent bending tests. Markers on the vertebrae were digitized through time (for digitizing methods, see McHenry et al. 1995), allowing us to measure the position of the segment’s rotational axis and to monitor the overlap of the articular process of the caudal vertebral body with the neural spine of the cranial vertebral body.

Dynamic bending tests were performed to measure dynamic mechanical properties and their dependence on angular deformation and angular frequency (see equations 2–4). To assess frequency-dependence, intervertebral joints were bent at bending frequencies of 0.5–5 Hz at 0.5 Hz increments (angular frequencies, ω, of π–10π rad s⁻¹). This range of bending frequencies was chosen to encompass the tailbeat frequencies (0.75–3 Hz) measured in live, steadily swimming bottlenose dolphins (Fish, 1993). It should be noted that the bottlenose dolphins were larger than the saddleback dolphins used in this study (2.61 m versus 2.06 m mean total length, respectively; Table 1); as a result, we may expect tailbeat frequencies in live saddleback dolphins to exceed 3 Hz. To assess strain-dependence, intervertebral joints were bent at five amplitudes: 0.012, 0.019, 0.026, 0.033 and 0.049 rad (0.6–2.9 °). This range of testing amplitudes was chosen to encompass the maximal intervertebral strains (2 °) calculated from a sustained locomotor sequence of a bottlenose dolphin (Pabst, 1993). It should be noted that tailbeat amplitude remains constant over a range of steady swimming speeds (Fish, 1993) and may increase during accelerations (Pabst, 1993). Because most of this motion is limited to the caudal tailstock, we measured dynamic properties only in the lumbo-caudal and caudal 7/8 segments. For each bending amplitude, intervertebral segments were bent over the entire range of bending frequencies.

To determine whether regional variation in vertebral morphology corresponded to regional variation in bending stiffness, we measured 15 linear characteristics of the vertebrae and intervertebral ligaments of all test segments (Fig. 4C,D; Table 2). These features were chosen because of their presumed mechanical importance in bending (see Discussion).

For quasi-static and dynamic bending tests, the dependence of stiffness, damping coefficient and resilience on the independent variables was determined using analysis of variance (ANOVA) (SAS Institute, 1985). Because we repeatedly measured mechanical properties from a single joint, and from many joints within an individual dolphin, we treated ‘individual’ as a factor (see Tables 3 and 7) that behaves like a randomized block effect without replication (Sokal and Rohlf, 1981). This removes the effects of differences between individuals from the other factors. For the quasi-static tests, main factors were individual, bending direction (‘direction’ in Table 3), structural disposition before and after cutting of ligaments (‘structure’), and axial joint position (‘joint’). In addition, interaction terms were also considered. Planned contrasts were run between categories within the structure and joint variables. For the dynamic tests, main factors were individual, joint, amplitude (‘amplitude’ in Table 7) and frequency (‘frequency’). Interaction terms were also considered. Planned contrasts were run within the amplitude and frequency variables. For correlations between morphological features and initial bending stiffness, we performed a stepwise linear regression to determine which morphological features were the best predictors (Wilkinson, 1989). We also measured the correlations among structural features using Pearson coefficients (r) and the Bonferroni procedure to correct for multiple comparisons (Wilkinson, 1989). For all analyses, the significance level was 0.05. Missing values are detailed in Table 1.

Quasi-static bending stiffness of the intervertebral ligaments varies regionally (Fig. 5; Tables 3, 4). In both extension and flexion, bending stiffness is greatest at the lumbo-caudal joint, with decreasing stiffness towards the head and tail. There are no significant differences between either thoracic 4/5 and thoraco-lumbar joint segments or caudal 7/8 and caudal 13/14 joint segments (Table 4). Initial stiffness is greater in extension than in flexion throughout the column (Table 3).

Cutting the interspinous ligaments significantly reduces the initial stiffness of the joints (Fig. 5; Tables 3, 4). Cutting the intertransverse ligaments has no detectable effect on stiffness. The interaction terms in the ANOVA are not significant (Table 3).

Vertebral motion as recorded on video tape showed that in all cases (two individuals, six joint positions, three structure treatments each) the joints were rotating about the intervertebral disc. Overlap of the articular processes of the neural spine of the caudal vertebra with the neural spine of the cranial vertebra, at low initial bending strains, was clearly seen in the intact and cut segments of thoracic 4/5, thoraco-lumbar, lumbo-caudal and caudal 7/8 (see also Fig. 1). The cranial neural spine of lumbar joint 10/11 showed movement of tissue during extension and flexion indicative of articular process overlap. No bony overlap was detected during cutting of the ligaments; it is possible, however, that a cartilaginous articular process is present on the intact caudal neural spine. No articular process overlap was detected for caudal joint 13/14. It is important to note that we did not detect bone-to-bone contact during the low-amplitude strains associated with the range of bending motions used in the measurement of initial stiffness.

While the sample size for the caudal 20/21 and prefluke/fluke joints does not permit statistical analysis (N=2 and 1, respectively; see Table 1), the most abrupt change in initial stiffness in the entire vertebral column, from approximately 25Nm rad⁻¹ to less than 1 N m rad⁻¹, was seen in these two neighboring joints (Fig. 6).

Using stepwise linear regression, six of the 14 morphometric variables (Table 2) are significantly correlated with initial bending stiffness in extension and flexion (Table 5). 55 % of the variation in initial stiffness during extension (r²=0.554) is predicted by the length and width of the intervertebral disc and the length of the cranial vertebral body in the segment (see Fig. 4 for morphology). 40 % of the variation in initial stiffness (r²=0.400) during flexion is predicted by the width of the nucleus pulposus, the length of the caudal vertebral body in the segment and the height of the transverse processes.

Other structural features are correlated with these structural predictors of initial bending stiffness (Table 6). Correlation analysis of all pairwise combinations of the 14 features yields the following significant relationships after correction for multiple comparisons using the Bonferroni procedure: (1) length of the intervertebral disc is negatively correlated with vertebral segment height (r=−0.622) and transverse process width (r=−0.653), (2) width of the intervertebral disc is positively correlated with articular process height (r=0.737), intervertebral disc height (r=0.919), annulus fibrosus dorsal height (r=0.661), nucleus pulposus width (r=0.716) and nucleus pulposus height (r=0.812), (3) length of the cranial vertebral body is positively correlated with the length of the caudal vertebral body (r=0.946), (4) width of the nucleus pulposus is positively correlated with intervertebral disc height (r=0.678) and intervertebral disc width (r=0.716) and (5) height of the transverse processes is negatively correlated with intervertebral disc height (r=−0.611).

When measured over a range of amplitudes and frequencies (θ₀ and ω, respectively, in equation 2), dynamic bending stiffness in the lumbo-caudal and caudal 7/8 joints varies as a function of amplitude only, increasing from 0.012 to 0.026 rad and remaining statistically constant at higher amplitudes (Table 7; Fig. 7A). In addition, the lumbo-caudal joint is significantly stiffer than the caudal 7/8 over the entire range of amplitudes (Table 7). Resilience also increases only with amplitude, increasing from an amplitude of 0.019 to 0.026 rad and then again from an amplitudes of 0.033 to 0.049 rad (Table 7; Fig. 7B). In addition, the lumbo-caudal joint has significantly greater resilience than does the caudal 7/8 joint over the entire range of amplitudes (Table 7).

When measured over a range of amplitudes and frequencies (see equation 2), the damping coefficient in the lumbo-caudal and caudal 7/8 joints decreases significantly with increasing bending frequency and increasing bending amplitude (Table 7; Fig. 7C,D). Damping coefficient decreases significantly only until an angular frequency of 4π rad s⁻¹ (2 Hz) is reached (Fig. 7C,D) and only begins to decrease with increasing bending amplitude at an amplitude of 0.026 rad.

Using mechanical tests, this study is the first to test the hypotheses that the bending stiffness of the dolphin’s vertebral column varies (1) with anatomical region, (2) with amplitude and frequency of joint motion and (3) with specific vertebral and intervertebral structures. We focus our discussion on these results and compare dolphin vertebral function with that of other aquatic vertebrates and terrestrial mammals.

In dorsoventral bending, the intervertebral joints of the saddleback dolphin are stiff near the middle of the body (i.e. the lumbo-caudal joint) and become more flexible towards the head and the tail (Fig. 5). The intervertebral joint at the base of the flukes is dramatically less stiff than any other joint tested in this study (Fig. 6).

Although, to the best of our knowledge, there are no kinematic data for swimming saddleback dolphins, the variation in the mechanical properties of their caudal vertebral column is consistent with variation in swimming motions measured from other species of odontocete cetaceans. For example, bottlenose dolphins swim by oscillating the caudal one-third of their bodies (Fish, 1993), where the caudal vertebral column displays decreased intervertebral joint stiffness compared with the lumbo-caudal joint (Fig. 5). During swimming in bottlenose dolphins, thrust is generated throughout the tailbeat cycle by continuously pitching the high-aspect-ratio fluke blades as they oscillate dorsoventrally (Fish, 1993). Mechanical results from our study suggest that the intervertebral joint at the base of the flukes acts like a low-resistance hinge, permitting subtle and continuous alterations of the angle of attack of the flukes.

Unlike the caudal tailstock, the thoracic region of bottlenose dolphins undergoes no measurable bending during steady swimming (Pabst, 1993). Mechanical results for saddleback dolphins, however, demonstrate that the thoracic and caudal intervertebral joints have similar stiffnesses (Fig. 5; Table 4). This lack of correspondence between intervertebral joint stiffness and observed regional mobility during swimming may be due to the removal of the ribs prior to mechanical testing. We hypothesize that the ribcage provides considerable bending resistance in vivo although, to our knowledge, this has not been tested.

The regional pattern of intervertebral joint stiffness is also consistent with the hypothesis that axial muscles anchored in the lumbar spine cause extension of the caudal tailstock (Pabst, 1993). The lumbo-caudal joint has the greatest initial bending stiffness, which suggests that it has the resistance to serve as a region of insertion for the robust epaxial (upstroke) muscles. Interestingly, the only epaxial muscle to insert onto the caudal vertebrae in the fluke, the m. extensor caudae lateralis, has the smallest force-generating capacity of any axial muscle. Pabst (1993) hypothesized that the movements of the flukes relative to the caudal tailstock must therefore be caused either by large hydrodynamic forces if the joint was stiff or by the smaller force available from the m. extensor caudae lateralis if the joint was flexible. The very low stiffness of the prefluke/fluke intervertebral joint suggests that the small forces generated by the m. extensor caudae lateralis may be sufficient to change or influence the angle of attack of the caudal fluke blades at times during the swimming cycle when hydrodynamic loads are small.

Intervertebral joint stiffness changes gradually along the length of the dolphin vertebral column (see Fig. 5; Table 4), except at the fluke vertebra, where there is a rather dramatic increase in joint compliance (Fig. 6). A similar pattern is seen at the lumbo-sacral joint of many terrestrial mammals (reviewed by Gal, 1993a). The lumbo-sacral joint has a large neutral zone (range of unloaded or unforced motion) and probably functions as a low-resistance joint to increase the range of movement of the pelvic girdle and hindlimb (Gal, 1993a).

The highly compliant fluke joint of dolphins and lumbo-sacral joint of terrestrial mammals are juxtaposed to joints of much higher stiffness. Thus, each intervertebral joint, as one in a series of discrete constructional units, represents an evolutionary ‘opportunity’ to modulate local mechanical behaviors along a continuous beam. Interestingly, we identified no morphological features of the caudal 20/21 versus prefluke/fluke vertebrae that would have allowed us to predict such abrupt differences in bending stiffness at the flukes (see below).

How does the regional pattern of intervertebral joint stiffness displayed in dolphins compare with that observed in other swimming vertebrates? Unlike dolphins, the caudal intervertebral joints of blue marlin are stiffer than the precaudal joints; the increased stiffness may increase the speed at which the undulatory propulsive wave travels down the body (Long, 1992). Similarly, in fish models, decreasing stiffness is associated with a decrease in the speed of the propulsive wave and a decrease in swimming speed (McHenry et al. 1995). Unlike undulatory swimming in fishes, dolphins are thunniform swimmers that do not use a traveling wave of bending to generate thrust. Thus, in dolphins, it is likely that stiffness functions in one of several other ways: (1) to provide localized resistance to bending (origin versus insertion) and (2) to decelerate the tail at the end of the stroke (resistance proportional to angular displacement). In powered flight, birds use their major flight muscles to accelerate and decelerate the wings (Dial, 1992). A stiff vertebral column might provide, without muscular input, the same function.

Dolphin intervertebral joints are stiffer in extension than they are in flexion (Fig. 5; Table 3). This is also true for a variety of terrestrial mammals, where the resistance to extension is caused by the touching of the articular facets of adjacent vertebrae (e.g. Gal, 1993b; Tencer et al. 1982). However, articular processes with opposing facets occur only in the cervical and cranial thoracic region in the vertebral column of bottlenose dolphins (Rommel, 1990). Thus, some other additional structural feature(s) must contribute to the stiffness in extension in the saddleback dolphin’s vertebral column (Fig. 5).

Resistance to both extension and flexion is dependent upon the integrity of the interspinous ligament (Fig. 5; Tables 3, 4). Resistance to extension is also positively correlated with width of the intervertebral disc and negatively correlated with the length of the intervertebral disc and length of the cranial vertebral body (r²=0.554). Resistance to flexion is positively correlated with the width of the nucleus pulposus and the height of the transverse process, and negatively correlated with the length of the caudal vertebral body (r²=0.400). These predictive structural features are also correlated with other vertebral and intervertebral features (Table 6).

How do these structures directly affect bending stiffness? We would predict, using simple beam mechanics (see Den Hartog, 1949; Wainwright et al. 1976; Denny, 1988), that the interspinous ligament would increase resistance to flexion by providing tensile structures distal to the axis of bending at the intervertebral disc (e.g. Slijper, 1946; Gal, 1993b). It is more difficult to understand how this ligament influences joint stiffness in extension, since connective tissue fibers forming the ligament would presumably slacken as the neural spines moved closer together. However, because our video tapes showed overlap of the articular processes of the caudal vertebrae with the neural spine of the cranial vertebrae in at least four of the six joint positions (see Results), we propose that a novel structural mechanism exists for the purpose of stiffening the joint in both extension and flexion (Fig. 8). This articular-process-stiffening mechanism works by placing ligaments in tension during both extension and flexion. As the articular processes of the caudal vertebra shear past the neural spine of the cranial vertebra, medio–laterally oriented ligaments are lengthened. The ligamentous anatomy of this proposed mechanism remains to be confirmed.

In addition, structural predictors of bending stiffness (Table 5) may be understood by examining their correlations with other structures (Table 6). For example, the correlation of increasing nucleus pulposus width with increasing stiffness in flexion is at first puzzling (Table 5); however, the dimensions of the nucleus pulposus width are positively correlated, in turn, with two other cross-sectional disc features: disc height and disc width (Table 6). Together, this correlated set of features increases with increasing bending stiffness in a manner consistent with beam theory (Denny, 1988). Using these and other statistically significant associations, we have designed two hypothetical vertebral segments, one flexible (lowest stiffness) and one stiff (highest stiffness) (Fig. 9). These hypothetical designs are amalgams of features from the vertebral segments examined with statistical sample sizes (Table 2, between thoracic 4/5 and caudal 13/14). The design for a flexible segment is characterized by a long intervertebral joint and vertebral body, short neural spines and lateral processes and a small-diameter intervertebral disc. The design for a stiff segment is characterized by a short intervertebral joint and vertebral body, long neural spines and lateral processes, and a large-diameter intervertebral disc. Flexible segments minimize stiffness with a long intervertebral disc, as would be predicted from beam theory, where deformation of a structure under a constant external load is proportional to the cube of the length of the structure (Denny, 1988). The small cross-sectional area of flexible segments further lowers the second moment of area of the structure, which is proportional to the flexural stiffness (Denny, 1988). The short neural spines also lower the overall second moment of area of the cross section of intervertebral ligaments. Finally, the long centra keep the articular processes on the neural spine from overlapping, thus inactivating the articular-process-stiffening mechanism described in the previous section. To understand the design of the stiff segments, these arguments are simply reversed. It is important to note that we do not understand the relationship between the length of the transverse processes, their vertical position and the segment’s bending stiffness. Transverse processes may serve other functions that are correlated with, but not causally related to, the bending mechanics of the vertebral segments.

These results underscore the view that the functional relationship between bending stiffness and vertebral anatomy is complex and species-specific (e.g. Gal, 1993a, b; Long, 1992; Parry, 1949; Shirazi-Adl, 1989; Slijper, 1946; Tencer et al. 1982). Soft tissues, such as the interspinous ligaments, determine, in part, the bending stiffness of the vertebral segments in saddleback dolphins. Furthermore, only a fraction of the variation in bending stiffness is predicted by skeletal features alone (Table 5). Thus, any attempt to infer changes in locomotor function, and hence evolutionary transformations, from skeletons alone must be undertaken cautiously.

The stiffness, resilience and damping behaviors of dolphin caudal intervertebral joints vary with bending amplitude (Fig. 7; Table 7). Intervertebral joint stiffness and resilience increase and damping coefficient decreases with increased bending amplitude. These patterns suggest that the dolphin caudal vertebral column is a stiff, moderately resilient beam when undergoing maximal bending. Although bottlenose dolphins do not appear to modulate tailbeat amplitude with increasing speeds during steady swimming (Fish, 1993), they do use maximal tail bending when accelerating from a near stop to speeds of approximately 2 m s⁻¹ (Wainwright et al. 1987; Pabst, 1993). Thus, amplitude-dependent mechanical behaviors of intervertebral joints may be more important during non-steady, accelerative swimming than during steady swimming in dolphins.

While measuring dynamic mechanical properties of elastic structures of the caudal tailstock at deformation frequencies of 2.2 Hz in Lagenorhynchus obliquidens, Bennett et al. (1987) and Blickhan and Cheng (1994) chose to ignore the bending properties of the vertebral column because of its low energy storage. That the maximal power required to bend a joint in our experiments varies over three orders of magnitude suggests that there may be circumstances in which the elastic properties of the vertebral column play a role in the mechanics of swimming. For example, at a bending amplitude of 0.05 rad (2.9 °) and a frequency of 10π rad s⁻¹ (5.0 Hz), caudal joint 7/8 achieves its maximal stiffness power relative to other amplitudes and frequencies (Fig. 10). Since the area under the power–time curve represents the mechanical work, the positive work under the stiffness–power curve represents the total energy available to power recoil. Since the cycle period is 0.2 s, and the time to bend the joint maximally is one-quarter of that period, integration of equation 5 from 0 to 0.05 s yields 0.05 J of elastic energy stored in the joint. If there are 20 joints in the caudal tailstock (see Fig. 1), and assuming for the moment that they are all identical, then a total of 1.0 J of elastic energy is available to help power the unbending of the body and thrust production. In the best case, where none of this energy is lost (a damping coefficient, c, of 0 kg m² rad⁻² s⁻¹), 1.0 J would contribute in different proportions to the total elastic energy thought to be available in the caudal region of three other dolphin species: 3 % for Lagenorhynchus oliquidens (34.06 J total), 7 % for Tursiops truncatus (13.46 J total) and 39 % for Sotalia guianensis (2.55 J total) (Bennet et al. 1987; Blickhan and Cheng, 1994). Anatomical data for Delphinus delphis are not available for the estimation of the total elastic energy. If elastic energy reduces the energy cost of locomotion in dolphins (see Blickhan and Cheng, 1994), any additional energy provided by the bending backbone would assist in that function.

Given this range of dynamic mechanical behavior (Fig. 10), how might the function of the caudal vertebral column differ with different swimming behaviors? For example, during steady swimming, dolphins increase speed by increasing tailbeat frequency (Fish, 1993). While little power is required to bend the vertebral column when using high-frequency, low-amplitude tailbeats, the proportion of energy recovered in the form of elastic recoil is also very low (approximately 20 %, see Fig. 7B). Thus, under these conditions, the vertebral column appears to act like a shock-absorber, not a spring. When using large-amplitude tailbeats, as might occur during accelerations, dolphins benefit from more energy return in the form of elastic recoil (approximately 50 %), but also require higher power output (2 W maximal) to bend the vertebral column at the fastest frequencies.

The resilience values of the dolphin intervertebral joints are lower than those reported for blue marlin (greater than 90 %; Long, 1992) and for a number of terrestrial mammals (65–75 % for lumbar joints in extension; Gal, 1993b). Why do dolphins possess a stiff but non-resilient vertebral column? The relatively low resilience of the dolphin intervertebral joints may provide dynamic stability to the column. That is, the joints may act to dissipate energy to prevent harmful break-away oscillations that may occur at frequencies near structural resonance (Meyhofer and Daniel, 1990). Alternatively, low resilience may simply be a consequence of increased vertebral flexibility. Gal (1993b) demonstrated that the lumbar intervertebral joints of tigers, jaguars and seals were less resilient (36–41 %) in flexion than those of monkeys or wallabies (65–67 %). Large cats and seals display great vertebral flexibility (e.g. Hildebrand, 1959; King, 1983).

The dynamic mechanical properties of dolphin intervertebral joints suggest that only at large bending amplitudes does the vertebral column store and release appreciable elastic strain energy. At lower amplitudes, the column dissipates strain energy with functional consequences, it is fair to say, that we do not yet completely understand.

Our results suggest that the pattern of body deformation observed in swimming dolphins is, in part, controlled by the mechanical behavior and morphology of the vertebral column. Clearly, these data are insufficient to explain dolphin swimming behavior completely. Factors that affect the pattern of whole-body deformation during swimming in vertebrates include (1) the mechanical properties of the axial skeleton (e.g. Long, 1992, 1995; Gal, 1993a,b; Bennett et al. 1987; Hebrank et al. 1990), (2) force production and distribution of the swimming muscles (e.g. Arkowitz and Rommel, 1985; Bennett et al. 1987; Blight, 1977; Long et al. 1994; Pabst, 1990, 1993; Rome et al. 1993; Wainwright, 1983; Westneat et al. 1993), (3) the mechanical properties of other body tissues that are deformed during swimming (i.e. fins, flukes, skin, subdermal sheaths) (Alexander, 1987; Fish and Hui, 1991; Pabst, 1996; Wainwright et al. 1978; Brainerd, 1994), and (4) interactions between the body and its fluid environment (Blight, 1977; Daniel and Webb, 1987; Fish, 1993; Gray, 1936; Lang, 1975; Lighthill, 1969; Vogel, 1994). Integrating data at all these levels is required to fully understand the mechanics of swimming in any vertebrate.

The authors wish to thank the following people for their contributions: Paul Bornemann, John Gosline, Ted Grand, Jim Mead, Charlie Potter, Butch Rommel, Meg Ronsheim, Steve Vogel and Steve Wainwright. This project was made possible through a cooperative agreement between the National Marine Fisheries Service and the Smithsonian Institution. J.H.L. and W.R.S. were supported by grant N000149310594 from the Office of Naval Research. D.A.P. and W.A.M. were supported by grant N000149310751 from the Office of Naval Research.