The 3-D model of the fish was created in Matlab using surfaces defined by X, Y, and Z matrices. The x-axis is along the length of the fish with the origin at the nose, y is the horizontal plane and z in the vertical. The profiles of the fish body are defined by the same coordinates used in the CAD model of a real tuna and NACA airfoil, and the caudal fin has a simple airfoil cross-section. The cross-sections of the body are ellipses defined by:

The sections on the tail and nose cone are equally spaced, and the cross-sections for the tail are defined by the rib locations. Using the time history of the spline, the position and orientation of each rib section and the tail can be determined. Transformation matrices are computed for each section of the moving tail and fin, pre-multiplied together, and then multiplied with the respective coordinates of the cross-sections.

Fig. 7. The tail components superimposed on the travelling wave
Fig. 8. The time step visualization of the tail motion

There are three main outputs of the spline motion and strain calculations, which can be run for various percentages of the tail period: a superimposed image of the tail position on the travelling wave, a visualization of the tail at each time step including the SMA wires, and graphs of the strain time histories for both side of the fish, shown in Figures 7 and 8, respectively. The circles in Figure 7 represent the locations of the axles on the tail. All figures are for 0.5s, at a frequency of 1Hz, with A0 equal to 0.5 (amplitude at tip of tail equal to 8cm), and k equal to 4.

Fig. 9. The SMA wire strain time histories

The e1 through £4 time histories illustrated in Figure 9 are the strains from the forward most section 1 through aft section 4, for the wires on both sides of the tail. Notice the SMA strains have a phase angle of 180° between opposing sides of the same section, and have the profile of sine waves as would be expected. The pre-strains in section one through four are: 0.04; 0.04; 0.035; 0.022.

4.4 The thermomechanical model derivation

In order to size the wires and controller power supply, a thermodynamic model of the heating and cooling of the wires was developed. The lumped capacitance model was used in a 1-D radial formulation, and the accompanying differential equation solved numerically using Matlab. The resistive heating of the wires was modelled based on a supplied current, and free convection was used for the heat transfer at the surface of the wires. The coating on the wires was neglected (since it was assumed to be a very thin film), as was the latent heat of phase transition. The latent heat was initially included in the model, but the small volume of the wires made the factor insignificant. The heat equation for heating of the wire was therefore derived as follows:

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