2D 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), 1
In two-dimensional problems, area moments of inertia apply to continual sections in bending and torsion 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 two-dimensional theories of moments of inertia apply to discrete data point sets 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, 2 for a double point, 3 for a triple point, etc.) instead of area (volume and/or mass in 3D problems). 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.
Show the essence of these theories by the simplest but most important linear approximation in the two-dimensional case by the following problem setting.
Given n (n ∈ N+ = {1, 2, ...}, n > 2) points [j=1n (x'j , y'j)] = {(x'1 , y'1), (x'2 , y'2), ... , (x'n , y'n)] with any real coordinates in a coordinate system x'O'y' . The first, or statical, moments are Sy' = Σj=1n x'j about the y'-axis and Sx' = Σj=1n y'j about the x'-axis. The second, or inertia, moments are Jy'y' = Σj=1n x'j2 about the y'-axis, Jy'x' = Σj=1n x'jy'j about the y'-axis and the x'-axis, and Jx'x' = Σj=1n y'j2 about the x'-axis. The x'-coordinate of the data center is x'c = Sy' / n = Σj=1n x'j / n , and the y'-coordinate of the data center is y'c = Sx' / n = Σj=1n y'j / n .
Now use centralization transformation x = x' - x'c = x' - Σj=1n x'j / n , y = y' - y'c = y' - Σj=1n y'j / n to provide xc = 0 and yc = 0 and hence coordinate system xOy central for the given data. In xOy , both Sy and Sx vanish. Further work in this system with points [j=1n (xj , yj)] to be approximated with a straight line ax + by = 0 containing origin O(0, 0). Here Sy = Σj=1n xj = 0, Sx = Σj=1n yj = 0, as well as (by the parallel axes theorem [8]) Jyy = Σj=1n xj2 = Jy'y' - nx'c2, Jyx = Σj=1n xjyj = Jy'x' - nx'cy'c , Jxx = Σj=1n yj2 = Jx'x' - ny'c2. Also consider another coordinate system XOY obtained from xOy by its turning (rotating) about their common origin O by any angle α positive in the anticlockwise direction so that we have axis OX from Ox and axis OY from Oy . Now X = x cos α + y sin α , Y = y cos α - x sin α , JYY = Jyy cos2 α + Jxx sin2 α + Jyx sin 2α , JYX = Jyx cos 2α - 1/2 Jyy sin 2α + 1/2 Jxx sin 2α , JXX = Jyy sin2 α + Jxx cos2 α - Jyx sin 2α . Further take declination angle α such that JYX = 0, i.e.
tan 2α = 2Jyx / (Jyy - Jxx).
This well known formula is not universal because it looses any sense by Jyy = Jxx . Naturally, by Jyy = Jxx and nonzero Jyx , we can consider 2α = π/2 and αX = π/4 with the remaining principal direction αY = 3π/4. But it is much better to obtain a general relation for determining α by any nonzero Jyx (otherwise, turning (rotating) is unnecessary at all).
We have
tan 2α = T = c/d (c = 2Jyx , d = Jyy - Jxx),
tan α = t ,
2t/(1 - t2) = T ,
Tt2 + 2t - T = 0 ,
t1 = [- 1 + (1 + T2)1/2] / T = d[- 1 + (1 + c2/d2)1/2] / c = [- d + (c2 + d2)1/2] / c ,
t1 = {Jxx - Jyy + [4Jyx2 + (Jyy - Jxx)2]1/2}/2/Jyx ,
t2 = [- 1 - (1 + T2)1/2] / T = d[- 1 - (1 + c2/d2)1/2] / c = [- d - (c2 + d2)1/2] / c ,
t2 = {Jxx - Jyy - [4Jyx2 + (Jyy - Jxx)2]1/2}/2/Jyx ,
α1 = arctan{{Jxx - Jyy + [4Jyx2 + (Jyy - Jxx)2]1/2}/2/Jyx},
α2 = arctan{{Jxx - Jyy - [4Jyx2 + (Jyy - Jxx)2]1/2}/2/Jyx}.
Then XOY becomes the principal central coordinate system with extremum values JXX of Jxx and JYY of Jyy , namely either JXX = (Jxx + Jyy)/2 + [(Jxx - Jyy)2/4 + Jyx2]1/2 , JYY = (Jxx + Jyy)/2 - [(Jxx - Jyy)2/4 + Jyx2]1/2, or vice versa. Also consider another coordinate system X'OY' obtained from XOY by its turning (rotating) about their common origin O by any angle α' positive in the anticlockwise direction so that we have axis OX' from OX and axis OY' from OY . Now JY'Y' = JYY cos2 α + JXX sin2 α , JY'X' = 1/2 JXX sin 2α - 1/2 JYY sin 2α , JX'X' = JYY sin2 α' + JXX cos2 α' . Introduce radii of inertia [8] ry = (Jyy/n)1/2, rx = (Jxx/n)1/2 and principal central radii of inertia rY = (JYY/n)1/2, rX = (JXX/n)1/2, place point rY in axis OX and point rX in axis OY , and obtain the ellipse of inertia [8].
Define and determine measures SL = (Jmin / Jmax)1/2 of data scatter and TL = 1 - SL = 1 - (Jmin / Jmax)1/2 of data trend with respect to linear approximation. Also by nonlinearity, S ≤ SL and T ≥ TL . Unlike the least square method, linear two-dimensional theories of moments of inertia provide best linear approximation to the given data, e.g. in numeric tests, see Figures 1, 2 with replacing (x’, y’) via (x , y):
Figure 1. SL = 0.218. TL = 0.782
Figure 2. SL = 0.507. TL = 0.493
These theories bring deep theoretical fundamentals for data estimation, approximation, and processing, namely as applied to the problems of the existence and uniquity of solutions, 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
[8] Pisarenko G., Yakovlev A., Matveev V. Aide-memoire de resistance des materiaux. - Moscou: Mir, 1985