Generalization of the super ellipsoid concept and its application in mechanics

Abstract

In this paper, we generalize the concept of super ellipsoid that was proposed by the French mathematician and mechanician Gabriel Lamé in 1818. Super ellipsoid itself represents a generalization of the ellipsoid, namely when the power n appearing in super ellipsoid equals 2, the conventional ellipsoid is obtained. In this paper the notion of super ellipsoid is further extended by introducing as many generally different powers, as is dictated by the dimensionality and the physics of the problem. Specifically, two different power parameters are introduced for the generalized super ellipse. Likewise, three generally different powers are adopted for the three-dimensional super ellipsoid etc. An applied mechanics problem is considered to demonstrate the efficiency of the notion of generalized super ellipsoids, namely we study the static behavior of an engine's crankshaft. Two to five geometric parameters specifying the crankshaft are treated as belonging to a generalized super ellipsoid set.

A contact binary satellite of the asteroid (152830) Dinkinesh

“…To produce a closed shape, and allow an estimation of the body volume, the unseen hemisphere has been approximated with an analytical solid figure. For this purpose, we chose a generalized super-ellipsoid 40 , whose implicit representation is given by the function in which x , y and z are the standard Cartesian coordinates. A fit to the reconstructed hemisphere leads to a = 0.40, b = 0.40, c = 0.35 km, k = m = 2 and n = 1.35.…”

“…To produce a closed shape, and allow an estimation of the body volume, the unseen hemisphere has been approximated with an analytical figure. For this purpose, we chose a generalized super-ellipsoid 40 , whose implicit representation is given by the function We estimated the uncertainty in the volume of Dinkinesh from the difference between the shape model and the super-ellipsoid convex shell. For the hemisphere covered by imaging, the difference in volume is 4.7%.…”