3D Inertia Moment Linear Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mechanical and Physical Journal
of the “Collegium” All World Academy of Sciences
Munich (Germany)
10 (2010), 2
In three-dimensional problems, volume and mass moments of inertia apply to rotating continual objects in mechanics.
In data processing, estimation, and approximation, the least square method [1] by Legendre and Gauss only usually applies to contradictory (e.g., overdetermined) problems. Overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], and data processing [6] have discovered a lot of principal shortcomings [7] of this method.
In these sciences, linear three-dimensional theories of moments of inertia are valid by coordinate system rotation invariance. By separate data points, use the quantity of everyone of them, e.g., its integer multiplicity (1 for a single point, for a double point, 3 for a triple point, etc.) instead of area, volume, and/or mass. By coordinate system translation invariance of the given data, centralize them by subtracting every coordinate of the data center from the corresponding coordinate of every data point.
Given n (n ∈ N+ = {1, 2, ...}, n > 3) points [j=1n (x'j , y'j , z'j)] = {(x'1 , y'1 , z'1), (x'2 , y'2 , z'2), ... , (x'n , y'n , z'n)] with any real coordinates in a coordinate system O'x'y'z' . The moments of inertia are Jx'x' = Σj=1n (y'j2 + z'j2) about the x'-axis, Jy'y' = Σj=1n (x'j2 + z'j2) about the y'-axis, Jz'z' = Σj=1n (x'j2 + y'j2) about the z'-axis, Jx'y' = Σj=1n x'jy'j about the x'-axis and the y'-axis, Jx'z' = Σj=1n x'jz'j about the x'-axis and the z'-axis, and Jy'z' = Σj=1n y'jz'j about the y'-axis and the z'-axis. For any u'-axis (containing O') whose angles with x'-axis, y'-axis, and z'-axis are α' , β' , and γ' , respectively, we obtain Ju'u' = Jx'x' cos2 α' + Jy'y' cos2 β' + Jz'z' cos2 γ' - 2Jx'y' cos α' cos β' - 2Jx'z' cos α' cos γ' - 2Jy'z' cos β' cos γ' . If u'-axis and v'-axis are obtained from x'-axis and y'-axis, respectively, via their turning (rotating) about their common origin O' by any angle θ positive in the anticlockwise direction about z'-axis, then Ju'v' = Σj=1n u'jv'j = Jx'y' cos 2θ + (Jx'x' - Jy'y')/2 sin 2θ . Use clearly invariant centralization transformation x = x' - Σj=1n x'j / n , y = y' - Σj=1n y'j / n , z = z' - Σj=1n z'j / n to provide coordinate system Oxyz central for the given data. Further work in this system with points [j=1n (xj , yj , zj)] to be approximated with a plane ax + by + cz = 0 containing origin O(0, 0, 0). The x'-coordinate of the data center is x'c = Σj=1n x'j / n , its y'-coordinate is y'c = Σj=1n y'j / n , and its z'-coordinate is z'c = Σj=1n z'j / n . By the parallel axes theorem, Jxx = Σj=1n (yj2 + zj2) = Jx'x' - n(y'c2 + z'c2), Jyy = Σj=1n (xj2 + zj2) = Jy'y' - n(x'c2 + z'c2), Jzz = Σj=1n (xj2 + yj2) = Jz'z' - n(x'c2 + y'c2), Jxy = Σj=1n xjyj = Jx'y' - nx'cy'c , Jxz = Σj=1n xjzj = Jx'z' - nx'cz'c , Jyz = Σj=1n yjzj = Jy'z' - ny'cz'c . For any Ou-axis whose angles with x-axis, y-axis, and z-axis are α , β , and γ , respectively, we obtain Juu = Jxx cos2 α + Jyy cos2 β + Jzz cos2 γ - 2Jxy cos α cos β - 2Jxz cos α cos γ - 2Jyz cos β cos γ . Take such α , β , and γ that Jxy = Jxz = Jyz = 0 to obtain a principal central coordinate system OXYZ . In Oxyz , the moment of inertia tensor is real and symmetric like a stress tensor. Then by the spectral theorem [1], there exists a Cartesian coordinate system OXYZ in which this tensor is diagonal with principal moments of inertia as diagonal elements JXX , JYY , and JZZ . If they all are distinct, then such a coordinate system OXYZ is unique. If precisely two of them coincide, then the third principal central axis is unique whereas the two principal central axes with coinciding principal moments of inertia can be simultaneously turned (rotated) about their common origin O (and the third principal central axis) by any angle. If all three principal moments of inertia coincide, then any central coordinate system is principal. For a principal central coordinate system OXYZ , Juu = JXX cos2 α + JYY cos2 β + JZZ cos2 γ . If one principal central axis, e.g., OZ , is known, then turn (rotate) the remaining two axes by an appropriate angle, e.g, φ in the case under consideration with tan 2φ = 2Jxy / (Jyy - Jxx). To obtain the ellipsoid of inertia, take values as radii of inertia aX = rYOZ = (Σj=1n Xj2 / n)1/2 in the X-axis, aY = rXOZ = (Σj=1n Yj2 / n)1/2 in the Y-axis, and aZ = rXOY = (Σj=1n Zj2 / n)1/2 in the Z-axis. The rectangular parallelepiped of inertia circumscribed about this ellipsoid of inertia has 8 vertices (corner points) (aX , aY , aZ), (aX , aY , - aZ), (aX , - aY , aZ), (aX , - aY , - aZ), (- aX , aY , aZ), (- aX , aY , - aZ), (- aX , - aY , aZ), and (- aX , - aY , - aZ) with coordinates in OXYZ . Then it is admissible to replace the given data point set with the set of these 8 points, each of them having quantity n/8, for calculating moments of inertia of the given data point set about any points, axes, and planes. If number n/8 is noninteger, then we have no problem because in overmathematics, quantities of elements in quantisets can be arbitrary [3].
These theories bring deep theoretical fundamentals for data estimation, approximation, and processing, namely as applied to the problems of the existence and uniqueness of solutions in data processing, estimation, and approximation, whereas least squared distance theories give explicit formulae more suitable.
Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.
References
[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994
[2] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416
[3] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[4] Lev Gelimson. Fundamental Science of Estimation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[5] Lev Gelimson. Fundamental Science of Approximation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[6] Lev Gelimson. Fundamental Science of Data Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[7] Lev Gelimson. Corrections and Generalizations of the Least Square Method. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 59-60