0
Research Papers

The Fractional Morphology and Growth Rate of the Nautilus pompilius: Preliminary Results Based on the R1-Fractional Trigonometry

[+] Author and Article Information
Carl F. Lorenzo

 NASA Glenn Research Center, Cleveland, OH 44135carl.f.lorenzo@nasa.gov

J. Comput. Nonlinear Dynam 6(2), 021004 (Oct 21, 2010) (10 pages) doi:10.1115/1.4002343 History: Received October 19, 2009; Revised March 05, 2010; Published October 21, 2010; Online October 21, 2010

This paper studies the morphology and evolutionary growth of the Nautilus pompilius based on the fractional R1-trigonometry. Morphological models based on the fractional trigonometry are shown to be superior to those of the commonly assumed logarithmic spiral. The R1-trigonometric functions further infer fractional differential equations, which, based on power law parametric functions, are used to develop a fractional growth equation modeling evolution from conception to maturity. An important aspect of this work is that it demonstrates a method of determination of the dynamic description of a fractional trigonometrically defined process from its morphological description.

FIGURES IN THIS ARTICLE
<>
Copyright © 2011 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

R1 Cosq,0(1,t) versus t time for q=0.2–1.0 in steps of 0.2

Grahic Jump Location
Figure 2

R1 Sinq,0(1,t) versus t time for q=0.2–1.0 in steps of 0.2

Grahic Jump Location
Figure 3

Phase plane plot, R1 Cosq,v(1,0,t) versus R1 Sinq,v(1,0,t) for a=1, v=0, and q=1–2 in steps of 0.1

Grahic Jump Location
Figure 4

Effect of v on an R1 spiral with a=1, q=1.2, and v=0.2 to −0.3 in steps of 0.1

Grahic Jump Location
Figure 5

Effect of a on R1-fractional spiral with q=1.2, v=0, and a=0.1–1.5 in steps of 0.2

Grahic Jump Location
Figure 6

Shell 2, morphological models: (a) logarithmic spiral, (b) R1-trigonometric spiral, (c) both spirals, and (d) both spirals enlarged (see Table 3 for parameters). Axes are in pixels.

Grahic Jump Location
Figure 7

Shell 3, morphological models: (a) logarithmic spiral, (b) R1-trigonometric spiral, (c) both spirals, and (d) both spirals enlarged (see Table 3 for parameters). Axes are in pixels.

Grahic Jump Location
Figure 8

Shell 4, morphological models: (a) logarithmic spiral, (b) R1-trigonometric spiral, (c) both spirals, and (d) both spirals enlarged (see Table 3 for parameters). Axes are in pixels.

Grahic Jump Location
Figure 9

Fractional spiral radius (pixels) and angle theta (50×rad) versus fractional spiral parameter tm. Shell 4.

Grahic Jump Location
Figure 10

x, y, and radius components of R1 trigonometric spiral shown in panel (b) of Fig. 8 versus fractional spiral parameter tm. Shell 4.

Grahic Jump Location
Figure 11

Spiral radii (in pixels) for logarithmic and fractional spirals versus spiral angle theta (in radians). Shell 4.

Grahic Jump Location
Figure 12

Expanded view spiral radii (in pixels) for logarithmic and fractional spirals versus spiral angle theta (in radians). Shell 4.

Grahic Jump Location
Figure 13

Shell 1, siphuncle morphological models: (a) image without model, (b) R1-trigonometric spiral of observed locus, (c) plate b enlarged, (d) spirals enlarged further with model of hidden locus indicated. See text and Table 4 for parameters. Axes are in pixels.

Grahic Jump Location
Figure 14

Growth rate in mm/day versus t time (days) (tm=kmtu=16.4t0.211). Shell 4.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In