This is the VMI mechanical fish, built on a frame and motor developed previuosly at Michigan State University. The body was developed by cadets Darren Wellner and Chris Petree '06.
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The animation above shows a depiction of a fish utilizing approximately 1000 points to describe a series of ellipses of various dimensions from the front to the rear of the fish. This fish consists of 59 ellipses, the dimensions of which are controlled by splines on the dorsal and lateral surfaces of the fish. Altogether, there are 6 degrees of freedom involved in determining the shape of the fish. With the shape defined, this model can be moved in any shape desired, such as the sinusoidal motion in the animation above.
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Many studies have been performed regarding self-propulsion of biological organisms. These include studies of all different sizes and types of creatures in both air and water media. The present research focuses only on neutrally buoyant organisms in an incompressible fluid. Considering only this portion of this spectrum of bio-mechanical self-propulsion, there are three major groups of cases which can be studied, as discussed by Fauci and Peskin [1]. Within the realm of large Reynolds numbers, several methods have been used in analyzing bio-mechanical propulsion. One of the first of these was set forth by Lighthill [2], which is a "small displacement" model.
Figure 1. Top view of a swimming fish, showing the coordinate system used. The origin is at the head of the fish and the y dimension is vertical (out of the page).
The equation for total average power required for the fish to move with a displacement h(x,t) is developed by Lighthill is
Likewise, the efficiency of the swimming body can be computed as the ratio of the amount of power available for propulsion to the total amount of power expended by the fish. Since the amount of power expended in propulsion is P = TU, the efficiency of swimming can be expressed as
Standing Wave Body Shape:
In order to calculate the equilibrium velocity U, or efficiency of any swimming motion, an assumption about the shape and nature of movement of the fish must be assumed. Several shapes were investigated as part of this research. The first shape investigated was that of a standing wave. Figure 2 shows examples of the shape of the fish for various values of c.
Figure 2. Three top views of the centerline of a swimming fish, each with a different value of c, the curvature coefficient.
Figure 3 shows a plot of the equilibrium velocity of the fish as a function of curvature coefficient for various values of amplitude, B. The velocity tends toward zero for c = p and c = 2p, but is a maximum at c = 3p /2. This corresponds with a zero slope at the end of the tail. The larger amplitudes bring about larger equilibrium velocities, however amplitudes larger than those shown here would be difficult to consider as "small". The velocities shown are for a frequency of oscillation of w = 1.
Figure 4 shows the swimming efficiency for the same parameters used in Figure 3. As with velocity, the efficiency is a maximum at c = 2p /3, corresponding to a zero slope on the tail at all times. It should be noted here that the amplitude of the movement of the body has no affect on efficiency since the amplitude cancels out of the numerator and denominator of the expression for efficiency.
Figure 3. Steady state velocity of the fish, as a function of curvature coefficient, at various amplitudes of oscillation.
Figure 4. Efficiency of swimming, as a function of curvature coefficient, at various amplitudes of oscillation.
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[1] L. Fauci and C. Peskin, “A Computational Model of Aquatic Animal Locomotion”, Journal of Computational Physics, Vol. 77, pp. 85-108, (1988).
[2] M. J. Lighthill, “Note on the Swimming of Slender Fish”, Journal of Fluid Mechanics, Vol. 9, pp. 305-317, (1960).
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