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 Lame 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.

[1]  C. Jiang,et al.  Non-probabilistic convex model process: A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems , 2014 .

[2]  K. M. Liew,et al.  Vibration and Buckling of Super Elliptical Plates , 1994 .

[3]  Panayiotis Papadopoulos,et al.  The modified super-ellipsoid yield criterion for human trabecular bone. , 2004, Journal of biomechanical engineering.

[4]  Barr,et al.  Superquadrics and Angle-Preserving Transformations , 1981, IEEE Computer Graphics and Applications.

[5]  Isaac Elishakoff,et al.  Interval, ellipsoidal, and super-ellipsoidal calculi for experimental and theoretical treatment of uncertainty: Which one ought to be preferred? , 2014 .

[6]  G. Altay,et al.  Static analysis of point-supported super-elliptical plates , 2008 .

[7]  Seyit Çeribaşı,et al.  Static analysis of superelliptical clamped plates by Galerkin's method , 2008 .

[8]  Philippe K. Zysset,et al.  Multi-axial mechanical properties of human trabecular bone , 2009, Biomechanics and modeling in mechanobiology.

[9]  I. Elishakoff,et al.  Convex models of uncertainty in applied mechanics , 1990 .

[10]  Isaac Elishakoff,et al.  Application of Lamé's Super Ellipsoids to Model Initial Imperfections , 2013 .

[11]  K. Liew,et al.  Vibrations of Perforated Plates with Rounded Corners , 1995 .

[12]  I. Elishakoff,et al.  Derivation of multi-dimensional ellipsoidal convex model for experimental data , 1996 .

[13]  K. M. Liew,et al.  Free Vibration Analysis of Thick Superelliptical Plates , 1998 .

[14]  Da-Guang Zhang,et al.  Non-linear bending analysis of super elliptical thin plates , 2013 .

[15]  Gen Yamada,et al.  Natural frequencies of square membrane and square plate with rounded corners , 1983 .

[16]  N. J. DeCapua,et al.  Transverse Vibration of a Class of Orthotropic Plates , 1972 .

[17]  Y. Ben-Haim.  Convex Models of Uncertainty in Radial Pulse Buckling of Shells , 1993 .