General Theory of Measuring Inhomogeneous Distributions
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)
9 (2009), 1
Measuring a physical magnitude not uniformly distributed in space and/or time [1] by means of a real physical device with nonzero sizes and inertia leads to inadequate measurement data. They are distorted (modulated) by some law and differ from the true distributions. The known measurement ways based on inadequate design methods cannot define the true distributions.
The purpose of the present theory is to reliably define the true distributionsof inhomogeneous magnitudes. This is achieved by designing and applying adequately modeling measurement processes. The distinctive feature of the proposed theory of measuring inhomogeneous distributionsis using new calculation formulae to determine the true distributions.
Let a one-variable function p(s) be continuously distributed and integral-mean modulated on each segment of a constant length Δ. Then the measurement operator to be inverted is [2-4]
MΔ: p(s) → p(s)=Δ-1∫s-Δ/2 s+Δ/2 p(t)dt.
The domain of p(s) can be restricted by not greater than Δfrom each boundary point. If p(s) and q(s) are solutions to this equation,
r(s) = q(s) - p(s)
is a solution to the equation [2-4]
∫s-Δ/2 s+Δ/2 r(t)dt = 0.
Its differentiating by s shows that for each s from the domain of definition, the equality
r(s + Δ/2) = r(s - Δ/2)
is an identity. Therefore, r(s) is a periodic function with period Δ. The integral mean value of this function on each segment of length Δvanishes. The inversion of MΔ is therefore ambiguous and determined up to such a function r(s), e.g., a finite or infinite sum of harmonic functions with divisors of Δ as periods. Let the device be such a suitable one that the variation of the distribution p(s) on each segment of length Δ is small enough in comparison with the least value of the distribution itself on the same segment. Then such a function r(s) can be considered vanishing and the inversion practically unambiguous. Otherwise, it is necessary and sufficient to additionally use another device with such constant Δ’ that Δ’/Δ is theoretically irrational and practically an uncancellable fraction with a sufficiently great sum of the numerator and denominator. If there is a continuous inversion of MΔ , then the function p(s) is continuously differentiable, and vice versa. By known discrete values of p(s) with measurement errors, interpolation and numerical differentiation lead to great errors. It is better to a priori choose standard functions whose linear combinations can be considered proper. When a priori choosing the coefficients by the arguments of those functions, due to the linearity of the measurement operator, we receive linear equations with unknown factors in the combinations, whose set is usually overdetermined and thus contradictory. The methods proposed by the author are very suitable for it and quickly lead to its quasisolutions. What is more, this brings no increasing but correcting measurement errors.
By many most important standard functions, using the measurement operator can be replaced with multiplication by a suitable factor. For example, this factor is 1 for each linear function,
sh(0.5nΔ)/(0.5nΔ) for exp(ns), sh ns, and ch ns;
sin(0.5nΔ)/(0.5nΔ) for sin ns and cos ns.
By a continuous function p(s) with period 2p, due to the uniform summability and hence integrability of the Fourier series, we have [2-4]
p(s) = 0.5c0 + Σn=1∞(cn cos ns + sn sin ns),
p(s) = 0.5c0 + Σn=1∞[sin(0.5nΔ)/(0.5nΔ)](cn cos ns + sn sin ns).
The inversion algorithm is hence determining the factors of p(s) and then the factors of p(s):
p(s) = 0.5c0 + Σn=1∞(cn cos ns + sn sin ns),
p(s) = 0.5c0 + Σn=1∞[0.5nΔ/sin(0.5nΔ)](cn cos ns + sn sin ns).
General theory of measuring inhomogeneous distributions applies to any problem in measurement technology and is especially useful and even urgent by very heterogeneous objects and rapidly changeable processes. A delay usually stable is simply compensated by a corresponding displacement of the measurement data in the positive direction of the time axis.
[1] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998
[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The “Collegium” All World Academy of Sciences Publishers, Munich, 2004
[3] Lev Gelimson. Providing Helicopter Fatigue strength: Unit Loads. 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, 589-600
[4] Lev Gelimson. Theory of Measuring Stress Concentration. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 53-54