Automatic control: the vertebral column of dogfish sharks behaves as a continuously variable transmission with smoothly shifting functions

Abstract

During swimming in dogfish sharks, Squalus acanthias, both the intervertebral joints and the vertebral centra undergo significant strain. To investigate this system, unique among vertebrates, we cyclically bent isolated segments of 10 vertebrae and nine joints. For the first time in the biomechanics of fish vertebral columns, we simultaneously characterized non-linear elasticity and viscosity throughout the bending oscillation, extending recently proposed techniques for large-amplitude oscillatory shear (LAOS) characterization to largeamplitude oscillatory bending (LAOB). The vertebral column segments behave as non-linear viscoelastic springs. Elastic properties dominate for all frequencies and curvatures tested, increasing as either variable increases. Non-linearities within a bending cycle are most in evidence at the highest frequency, 2.0 Hz, and curvature, 5 m −1. Viscous bending properties are greatest at low frequencies and high curvatures, with non-linear effects occurring at all frequencies and curvatures. The range of mechanical behaviors includes that of springs and brakes, with smooth transitions between them that allow for continuously variable power transmission by the vertebral column to assist in the mechanics of undulatory propulsion.

Mechanical behavior of shark vertebral centra at biologically relevant strains

“…Variations in body wavelength are likely influenced by vertebral column mechanical behavior (Porter et al, 2014;Donley et al, 2004;Long et al, 1994). In the cartilaginous vertebral column of sharks, strain (structural deformation) occurs not only at the intervertebral joints but also within individual centra, allowing the entire vertebral column to engage as a spring at high tailbeat frequencies and shift into a brake at low tailbeat frequencies (Porter et al, 2014(Porter et al, , 2016. The combination of strain occurring both within centra and at intervertebral joints results in greater total deformation along the vertebral column in cartilaginous fish when compared with their bony fish counterparts, and allows for greater elastic energy storage (Porter et al, 2014).…”

“…Previously considered a flimsier skeletal material than bone, mineralized cartilage of the vertebral column not only meets the mechanical demands of undulating sharks but also serves as a lighter alternative (Porter et al, 2006(Porter et al, , 2007(Porter et al, , 2014(Porter et al, , 2016Long et al, 2011;Porter and Long, 2010;Vogel, 1988). Mechanical properties of the shark vertebral column vary within individuals, between individuals and among species (Porter et al, 2006(Porter et al, , 2007.…”

Regional variation in undulatory kinematics of two hammerhead species: the bonnethead (Sphyrna tiburo) and the scalloped hammerhead (Sphyrna lewini)

“…Finally, the double oscillation system shown here, in which the anterior body undulates at a different frequency than the posterior body, may be specific to cartilaginous fishes (Long, 1995). Shark vertebral columns are non-linear, viscoelastic systems that store and transmit energy, behaving as both a spring and a brake, which allows for the differential transmission of power depending on the undulatory amplitude and frequency (Porter et al, 2014(Porter et al, , 2016. We hypothesize that the variable mechanical behavior of the cartilaginous vertebral column allows for high-frequency undulation in the anterior body without disrupting lower-frequency, posterior body undulation.…”

“…These complexities in undulatory wave propagation have also been observed in the lamprey and eel, and they may stand apart from the traditional rigid-bodied teleost model, where undulatory amplitude increases rostro-caudally (Long, 1995;Root et al, 1999;Long et al, 2010). With the exception of the eel, these patterns have been observed in fishes with cartilaginous vertebral columns or notochords, and are perhaps due to the mechanical properties of cartilaginous fishes (Porter et al, 2014(Porter et al, , 2016Long et al, 2002Long et al, , 2004. It has been suggested that anterior body flexion may be a result of recoil from body undulation, or that it may control the driving frequency of undulation; however, recent studies show that movement of fishes' heads increases their sensitivity to external stimuli (McHenry et al, 1995;Webb, 1988, Akanyeti et al, 2016.…”

Long axis twisting during locomotion of elongate fishes

“…If torsional damping differs along the length of the body, this could cause wobble amplitude to vary along the body. Long et al (2002) and Porter et al (2016) measured the flexural elastic and damping properties of lamprey bodies and dogfish vertebral columns, respectively, but only on one point along the body. These properties are not known for torsion, nor is it known how they vary along the body.…”