Providing Helicopter Fatigue Strength: Flight Conditions [Overmathematics and Other Fundamental Mathematical Sciences]

by

© Ph. D. & Dr. Sc. Lev Gelimson

RUAG Aerospace Services GmbH, Germany

Key words: Configuration, condition, elastic mathematics, reserve, reliability, fatigue.

Abstract. Due to fuel consumption by flight conditions, typical helicopter configurations have nonstationary distributions of pitch, roll, and yaw moments, forces, and stresses. For such analysis, test data scatter, and many other typical problems, classical mathematics has no concepts and methods adequate and general enough. This holds for the real numbers, sets, cardinalities, measures, probabilities, absolute and relative errors, as well as for the least square method, reliability, and risk estimations. There is no possibility to invariantly estimate the contradictoriness of a problem and the confidence in the exactness of an exact solution. The last problem also needs certain generalizations along with separation methods. Elastic mathematics provides many general concepts and methods fully adequate and very effective by estimating approximation quality, exactness confidence, reliability, and risk. All the obtained results apply to data scatter analysis, predicting the gravity center movement, and improving the fatigue strengths of responsible structural elements, e.g., in aircraft building.

1. INTRODUCTION

A helicopter has different configurations and has to be designed for certain conditions [1]. Its structural elements with distributions of moments, shears, displacements, and stresses need sufficient static and fatigue strengths [2 - 6]. By consuming fuel during flight, the helicopter center of gravity changes its relative position with influence on those distributions. Also for test data scatter and many other typical problems [2 - 6], there are no adequate tools in classical mathematics [7]. The real numbers evaluate not every bounded quantity; the sets, fuzzy sets, multisets, and set operations express not all collections; the cardinalities, measures, and probabilities are not sufficiently sensitive to infinite sets and even to intersecting finite sets. The absolute error is not sufficient for approximation quality estimation. The relative error is uncertain in principle and has an applicability domain very restricted. The unique known method applicable to overdetermined problems usual in data processing is the least square method with narrow applicability and adequacy domains and many fundamental defects. No known proposition applies to estimating the quality of approximations to distributions. Even artificial simplification by randomizing parameters in deterministic problems to apply stochastic approaches to estimating risk in reliability theory etc. brings complicated formulae.

Elastic mathematics [8 - 10] by the author provides many new concepts and methods very suitable for setting and solving these and a lot of other typical problems. Quantianalysis [11, 12] leads to universal quantisets, multiquantities, and uninumbers reliably modeling and evaluating any objects. Also by distributions, the introduced definitions and determinations of unierrors and reserves [13, 14] bring adequately estimating approximation quality, exactness confidence, reliability, and risk. The iteration methods of the least normed powers [15], of the unierror and reserve equalization [15], and of the direct solution [10] give both quasisolutions to general problems even with contradictoriness and for the first time its invariant measure. This improves data scatter analysis, predicting the gravity center movement, and the fatigue strengths of responsible structural elements in aircraft building [2, 3, 16], e.g., in a helicopter.

2. SOME HELICOPTER DESIGN PROBLEMS

2.1. Configurations and conditions

A helicopter with some typical configurations should be reliable by flight conditions such as vertical take-off, maximum speed, pushover, yawing, symmetric and rolling pullout1.

2.2. Moving the center of gravity

During flying, the helicopter center of gravity changes its position due to fuel consumption.

2.3. Test data scatter

Excluding random deviations by measurement needs properly analyzing test data scatter.

3. RELEVANT CONCEPTS AND METHODS OF CLASSICAL MATHEMATICS

3.1. Mixed magnitudes, sets, measures, probabilities, cardinal and real numbers

For concrete (mixed) quantities, e.g., “100 liter fuel’’, there is no suitable mathematical model and no known operation, say between “100 liter’’ and “fuel’’ (neither “100 liter’’ × “fuel’’ nor “fuel’’ × “100 liter’’). Set operations with absorption by both finite and infinite sets are very restrictedly reversible and, therefore, hinder constructing any universal degree of quantity.

The sets with either unit or zero quantities of their possible elements, the fuzzy sets with intermediate quantities in the indeterminate case only, and the multisets whose element quantities are any cardinal numbers [7] cannot express many collections even by element quantities between 0 and 1, e.g., half a fuel reserve and a quarter of a water reserve.

The measures cannot discriminate the empty set ∅ and null sets [7]; probabilities – impossible events and many typical events differently possible. For uncountable sets of elementary events, also uncountable sums of their probabilities should be considered.

The cardinality [7] is sensitive to finite unions of disjoint finite sets only; each measure is finitely sensitive within a certain dimension; they give the same continuum cardinality c and the same measure (either 0 or +∞), respectively, for very distinct sets of points between two parallel lines or planes differently distant from one another.

The real numbers R evaluate no unbounded quantity and, because of gaps, not all bounded ones. For example, the same probability p_n = p of the random sampling of a certain number

n∈N = {0, 1, 2, ...}

(N the natural numbers) does not exist in R, since the countable sum (which should be 1)

Σ_n∈N p_n

vanishes for p = 0 and is +∞ for p > 0. It is urgent to exactly express, in some suitable extension of R, all infinite and infinitesimal quantities, e.g., such a p for any countable or uncountable set, or distributions and distribution functions on sets of infinite measure.

3.2. Errors, reliability, and risk

The absolute error [7] alone offers no sufficient approximation quality estimation and gives, for example, the same result 1 for the formal (either true or false, i.e. either correct or incorrect) equality 1000 =? 999 acceptable in many cases as an approximate equality and for the inadmissible one 1 =? 0. The absolute error is not invariant by equivalent transformations of a problem because, for instance, when multiplying a formal equality by a nonzero number, the absolute error is multiplied by the modulus (absolute value) of that number.

Even by a general formal equality a =? b of two numbers a and b, there are two propositions

δ_{L , a =? b} = |a - b|/|a|,

δ_{R , a =? b} = |a - b|/|b|

to define the relative error [7] (left and right, respectively). This uncertainty generally inadmissible could be acceptable only if the ratio |a|/|b| is close to 1. The relative error is so intended that it should always belong to the closed interval [0, 1]. But for the formal equality 1 =? 0 by choosing 0 as the denominator, the relative error is +∞, for 1 =? -1 by each denominator choice it is 2 and, generally, has a restricted applicability domain amounting to formal equalities of two elements only whose moduli ratio can be considered close to 1. By more complicated formal equalities with at least three elements, e.g., by 100 - 99 =? 0 or 1 - 2 + 3 - 4 =? -1, a denominator choice seems to be vague at all. That is why the relative error is commonly used by the simplest equalities of two numbers only and very seldom by variables and functions.

There is no possibility in classical mathematics [7] to discriminate exact objects or models by the confidence in (reliability of) their exactness. This holds, e.g., for the following inequation and two of its exact solutions (x₁ practically unreliable and x₂ guaranteed):

x > 1, x₁ = 1 + 10^-10, x₂ = 1 + 10¹⁰. (1)

To estimate reliability and risk, a stochastic approach is commonly used [7]. To apply it to deterministic problems, their parameters are often artificially randomized. The corresponding distributions are considered known (even if they are really unknown) and most suitable for calculation. But even this simplification brings complicated formulae and analysis difficulties.

3.3. Least square method

The least square method [7] (LSM) is based on the absolute error alone and ignores the noncoinciding physical dimensions (units) of relations in a problem. The method does not correlate the deviations of the desired approximations from the approximated objects with these objects themselves, simply mixes those deviations without their adequate weighing, and considers equal changes of the squares of those deviations with relatively less and greater moduli (absolute values) as equivalent ones. The method foresees no iterating, is based on a fixed algorithm accepting no a priori flexibility, provides no own a posteriori adapting, and uses no invariant approximation quality estimation. These defects in the essence lead to many fundamental shortcomings in the applicability of the method. It loses any sense and is not applicable to problems simulated by a set of equations with different physical dimensions (units). The result of the method has no objective sense and is not invariant by equivalent transformations of a problem, e.g. by three equivalent sets of two equations (in parentheses)

x = 3/2 (x = 1, x = 2),

x = 102/101 (10x = 10, x = 2),

x = 201/101 (x = 1, 10x = 20).

Hence the method restricts the class of acceptable equivalent transformations and ignores equations with less factors. For less values, the method paradoxically brings greater (even absolute) errors and, in turn, for greater values, less (even absolute) errors; by relative errors and unierrors [10] such paradoxicality is still much stronger. Fitting two points in the plane x0y

x₁ = 1, y₁ = 1; x₂ = 10, y₂ = 15 (2)

by y = kx (k required) brings k = 151/101. For the first point (2) with less coordinates, the absolute error, left and right relative ones, and unierror [13, 14] are paradoxically much greater:

Δ_{k =? 1} = |k - 1| = 51/101,

δ_{L , k =? 1} = |k - 1|/|k| = 51/151,

δ_{R , k =? 1} = |k - 1|/|1| = 51/101,

δ_{k =? 1} = |k - 1|/(|k| + |1|) = 50/252 = 25/126,

respectively, than for the second point (2) with relatively greater coordinates:

Δ_{10k =? 15} = |10k - 15| = 5/101,

δ_{L , 10k =? 15} = |10k - 15|/|10k| = 5/1510 = 1/302,

δ_{R , 10k =? 15} = |10k - 15|/|15| = 5/1515 = 1/303,

δ_{10k =? 15} = |10k - 15|/(|10k| + |15|) = 5/3025 = 1/605.

The deviations of the obtained result from the approximated objects are not analyzed by the method at all. Each result is presented by the method as the highest truth without any justification. The obtained results cannot be refined by the least square method itself.

4. RELEVANT CONCEPTS AND METHODS OF ELASIC MATHEMATICS

4.1. Quantianalysis: Quantielements, quantisets, multiquantities, and uninumbers

A set quantioperation (quantirelation) [10, 11] is a set operation (relation) such that the actual quantity of each element of its operands (objects) is exactly taken into account, and can be denoted by the sign of a similar usual set operation (relation, respectively) with a little circle on the right above if the operation and the quantioperation give different results.

A quantiset [10, 11] is a nonpositional quantiunion of quantielements [10, 11] of the form _qa , each of them consisting of its element (basis), a , with its own quantity (named: uniquantity), q , inside in the quantiset, the elements and element quantities being any, possibly fuzzy [7], objects:

A =° {... , _qa , ... , _rb , ... , _sc , ...}° =° ... ∪° _qa ∪° ... ∪° _rb ∪° ... ∪° _sc ∪° ... .

Quantifying is a set quantioperation q: a → _qa . The empty quantielement of forms ₀a =° _q# (# the empty element, # ∈ ∅) is the empty set [7] ∅ and should be reduced to canonical form ₀#. The quantielements with a common basis should be reduced (collected) by adding quantities:

... ∪° _qa ∪°... ∪° _ra ∪° ... ∪° _sa ∪°... =° _{... + q + ... + r + ... + s + ...} a .

Quantisets are quantiequal if, after the reduction, they contain all quantielements in common. Outside quantifying a quantiset means multiplying the inside quantities by the outside one:

_tA =° _t{... , _qa , ... , _rb , ... , _sc , ...}° =° {... , _tqa , ... , _trb , ... , _tsc , ...}°.

{₂pilot, ₇passenger, _{100 kg}cargo, _{300 km}distance, _{-1 h}time, _{-300 liter}fuel}

is, e.g., a possible result of a helicopter flight. The multiquantity [10, 11] of a quantiset is the quantisum [10, 11] of the own (inside) quantities (uniquantities) of the quantiset elements (bases):

Q(A) = Q{... , _qa , ... , _rb , ... , _sc , ...}° = ...+ q +...+ r +...+ s +... .

The multiquantities of some chosen canonical sets coincide with their cardinalities [7], e.g.:

Q{a} = 1, Q[0, 1[ = c ,

Q(Z) = 2ℵ instead of ℵ₀

for any object a, the interval [0, 1[ including 0 but excluding 1, and the set of the integers Z .

To extend and refine the scale of the real numbers, they along with the infinite cardinal numbers [7] by preserving the properties of the usual operations and relations [7] build the uninumbers [10, 11]. For the same probability p_n = p of the random sampling of a certain n∈N ,

Σ_n∈N p = 1, Q(N) × p = 1, p = 1/Q(N) = 1/(ℵ + 1/2).

4.2. Unierror, reserve, reliability, and risk

For a formal number equality a =? b, the linear and quadratic unierrors always belonging to interval [0, 1] and irreproachably correcting and generalizing the relative error are defined as

δ_{a =? b} = |a - b|//(|a| + |b|) [a//b = a/b by a ≠ 0; a//b = 0 by a = 0 and any b],

²δ_{a =? b} = |a - b|//[2(a² + b²)]^1/2:

δ_{0 =? 0} = 0; ²δ_{0 =? 0} = 0;

δ_{1 =? 0} = 1; ²δ_{1 =? 0} = 1/2^1/2;

δ_{1 =? -1} = 1; ²δ_{1 =? -1} = 1;

δ_{100 =? 99} = 1/199.

For a formal vector equality Σ_ω∈Ω z_ω =? 0, the linear and quadratic unierrors are

δ(Σ_ω∈Ω z_ω =? 0) = ||Σ_ω∈Ω z_ω||//Σ_ω∈Ω ||z_ω||,

²δ(Σ_ω∈Ω z_ω =? 0) = ||Σ_ω∈Ω z_ω||//(Q(Ω)Σ_ω∈Ω ||z_ω||²)^1/2

where Q(Ω) is the multiquantity [10, 11] of the index quantiset Ω, and we have, e.g.:

δ_{100 - 99 =? 0} = 1/199 = δ_{100 =? 99};

δ_{1 - 2 + 3 - 4 =? -1}= |1 - 2 + 3 - 4 + 1|/(1 + 2 + 3 + 4 + 1) = 1/11.

For the equality with index λ in a quantiset of equations over indexed functions

_w(λ)(L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]] = 0) (λ∈Λ) (3)

where

L_λ is an operator with index λ from an index set Λ ;

f_φ is a function (dependent variable) with index φ from an index set Φ ;

z_ω is an independent variable with index ω from an index set Ω ,

the unierror is defined and determined as an estimating fraction (by z_λ’ → z_λ ; V volume)

δ_λ(m(λ)) = {lim [V(z_λ’)]^-1∫(||L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]]||_λ//sup||L_λ[_φ∈Φ f_φ’[_ω∈Ω z_ω]]||_λ)^m(λ)dV(z_λ’)}^1/m(λ)

where m(λ) is a positive number, we shall simply take 1;

in the denominator, a direct (not composite) function of independent variables is used and by determining the least upper bound, all different isometric transformations (conserving norms)

||f_φ’[_ω∈Ω z_ω]||_φ = ||f_φ[_ω∈Ω z_ω]||_φ

of even equal elements are considered. For the complete quantiset (3) of the equalities (n > 0),

ⁿδ(m) = {Σ_λ∈Λ w(λ)[δ_λ(m)]ⁿ//Σ_λ∈Λ w(λ)}^1/n.

The reserve R ∈ [-1, 1] of any inexact object is defined as its unierror with the opposite sign. For an exact object, suitably map the object with respect to its exactness boundary and simply take the unierror of the mapped object. For inequalities, replace such a mapping by negating inequality relations and conserving equality ones. For the inequation and its solutions (1),

R_{x > 1}(x₁) = R_{x > 1}(1 + 10^-10) = δ_{x <? 1}(1 + 10^-10) = 10^-10/(2 + 10^-10),

R_{x > 1}(x₂) = R_{x > 1}(1 + 10¹⁰) = δ_{x <? 1}(1 + 10¹⁰) = 10¹⁰/(2 + 10¹⁰).

Let a general problem [14] be a quantisystem [10] of quantirelations [10] containing both known elements (regarded as values) and unknown ones (regarded as variables).

A pseudosolution to such a problem is an arbitrary quantisystem of such values of all the corresponding variables that, after replacing each variable with its value, the problem becomes a quantisystem of quantirelations which contain known elements only and become determinable (either true or false). The pseudosolution is the set of all pseudosolutions.

If each of the last quantirelations is true, then such a pseudosolution is also called a solution to the problem. The solution is the set of all solutions to the problem.

If a pseudosolution to such a problem has the least unierror or the greatest reserve, respectively, by a specific realization of a certain method to define and determine them among all pseudosolutions to the problem, then this pseudosolution is called a quasisolution to the problem. A quasisolution is not necessarily a solution, which is especially important in contradictory problems that have no solutions in principle but can possess quasisolutions. The quasisolution is the set of all quasisolutions to the problem by this realization of that method.

If a solution to such a problem has the greatest reserve (note that all solution unierrors vanish) by a specific realization of a certain method among all solutions to the problem, this solution is called a supersolution to the problem. A supersolution to such a problem is not necessarily its quasisolution because the set of all solutions is a subset of the set of all pseudosolutions. If the both exist, then a quasisolution, which is not necessarily a solution, has a not less reserve in comparison with a supersolution. If in the last comparison, namely the strict inequality holds, then a quasisolution is certainly no solution. The supersolution is the set of all supersolutions to the problem by this realization of that method.

If a pseudosolution to such a problem has the greatest unierror or the least reserve, respectively, by a specific realization of a certain method among all pseudosolutions to the problem, then this pseudosolution is called an antisolution to the problem. The antisolution is the set of all antisolutions to the problem by this realization of that method.

Quasisolutions and supersolutions, as well as antisolutions, not necessarily exist because a set of unierrors or reserves not necessarily contains its greatest lower bound and least upper one.

The only element of the supersolution to the compound inequation 1 < x < 2 is x = 2^1/2 due to

R_{x > 1}(x) = R_{x < 2}(x),

δ_{x <? 1} = δ_{x >? 2} ,

|x - 1|/(|x| + |1|) = |x - 2|/(|x| + |2|),

(x - 1)/(x + 1) = -(x - 2)/(x + 2).

The supersolution to the compound inequation a < x < b contains the only element

x = (ab)^1/2, x = - (ab)^1/2, x = 0,

and is empty by a > 0, b < 0, a < 0 < b, and ab = 0, respectively, as well as to the overdetermined set

x = a, x = b. (4)

The reliability S, e.g., of solutions x₁ and x₂ to inequality (1), is defined and determined via reserve R as

S = (1 + R)/2:

S_{x > 1}(x₁) = S_{x > 1}(1 + 10^-10) = (1 + R_{x > 1}(1 + 10^-10))/2 = (1 + 10^-10)/(2 + 10^-10),

S_{x > 1}(x₂) = S_{x > 1}(1 + 10¹⁰) = (1 + R_{x > 1}(1 + 10¹⁰))/2 = (1 + 10¹⁰)/(2 + 10¹⁰).

The risk r, e.g., of solutions x₁ and x₂ to inequality (1), is defined and determined via reliability S as

r = 1 - S:

r_{x > 1}(x₁) = r_{x > 1}(1 + 10^-10) = (1 - R_{x > 1}(1 + 10^-10))/2 = 1/(2 + 10^-10),

r_{x > 1}(x₂) = r_{x > 1}(1 + 10¹⁰) = (1 - R_{x > 1}(1 + 10¹⁰))/2 = 1/(2 + 10¹⁰).

4.3. Unierror and reserve iteration methods of the least normed powers

The unierror method (LNPM) minimizes the mean value (by z_λ’ → z_λ ; V volume)

δ(f^(k), f^(k+1), g) = {[Σ_λ∈Λ w(λ)]^-1Σ_λ∈Λ w(λ)(lim [V(z_λ’)]^-1∫(||L_λ[_φ∈Φ f_φ^(k+1)[_ω∈Ω z_ω]]||_λ//sup||L_λ[_φ∈Φ f_φ’[_ω∈Ω z_ω]]||_λ)^gdV(z_λ’)}^1/g

of the unierror of a pseudosolution to the quantiset (3) of equations where

g is a positive number that is 2 in the method of the least normed squares (LNSM);

f^(k+1), f^(k) are the (k+1)st and kth approximations to a quasisolution to equations quantiset (3);

in the denominator, a direct (not composite) function of independent variables stands, and by determining the least upper bound, all different isometric transformations (conserving norms)

||f_φ’[_ω∈Ω z_ω]||_φ = ||f_φ^(k)[_ω∈Ω z_ω]||_φ

even of equal elements are used. This ensures that each next approximation to a quasisolution to the quantiset of equations (3) can be expressed via the prior approximation already known, which provides iterating. Alternatively (especially if set Λ is infinite) minimize the unierror

δ(f^(k), f^(k+1)) = sup_λ∈Λlim [V(z_λ’)]^-1∫(||L_λ[_φ∈Φ f_φ^(k+1)[_ω∈Ω z_ω]]||λ//sup||L_λ[_φ∈Φ f_φ’[_ω∈Ω z_ω]]||_λ)dV(z_λ’) (z_λ’ → z_λ)

where sup M is the least upper quantibound on an ordered quantiset M , i.e. the quantiset of the least upper bounds on the subsets of M reduced from above. The least upper quantibounds on two quantisets are ordered by ordering the usual least upper bounds on their quantisubsets minimally equally reduced from above to discriminate them. Replacing the unierrors with the reserves that naturally should be maximized brings the reserve method as a further method generalization also applicable to determining the supersolution to a set of relations.

4.4. Unierror and reserve equalizing iteration methods

A unierror equalizing iteration method (EEM) is still more effective. For a pseudosolution

[_φ∈Φ f_φ[_ω∈Ω z_ω]]

at any point [_ω∈Ω z_ω], the unierror of each (λth) equation in the quantiset (3) is determined:

δ_λ[_ω∈Ω z_ω] = ||L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]]||_λ//sup||L_λ[_φ∈Φ f_φ’[_ω∈Ω z_ω]]||_λ .

Now the λth equation (3) can be equivalently transformed with clear mathematical sense:

(L_λ[_φ∈Φ f_φ^(k+1)[_ω∈Ω z_ω]]//sup||L_λ[_φ∈Φ f_φ’[_ω∈Ω z_ω]]||_λ) ×

(||L_λ[_φ∈Φ f_φ^(k)[_ω∈Ω z_ω]]||_λ//L_λ[_φ∈Φ f_φ^(k)[_ω∈Ω z_ω]]) = 0 (λ∈Λ)

where the second fraction drops out if its denominator vanishes. If the kth approximation were substituted for the (k+1)st one, the left-hand side of the last equation would be equal to the unierror δ_λ(k) of the kth approximation. Order the quantiset of the unierrors δ_λ(k). The left-hand sides of two equations with the greatest modulus of the difference of their unierrors δ_λ(k) with corresponding indexes λ are setting equal to each other. Repeat such a procedure for the remaining equations, and so on. By every step of designing such an equalizing quantiset of equations, each initial equation is used at most once if possible. A new equation drops out if it can be derived from the already designed equations of the equalizing quantiset. Independently of ending each step, this designing is finished when the equalizing quantiset of equations has exactly one solution. If this quantiset contains all possible independent new equations and nevertheless has more than one solution, design the required amount of additional new equations that express equalizing the greatest unierrors δ_λ(k) to zero. Replacing the unierrors to be minimized with the corresponding reserves to be maximized leads to a reserve equalizing iteration method generalizing the described unierror equalizing iteration method and additionally allowing supersolutions determination. These methods are very effective by solving problems simulated by contradictory quantisets of equations, e.g., by data scatter. In the above problem (2), unlike the least square method (LSM), see Figure 1, the least normed square method (LNSM) and the EEM give an adequate result via the condition

|k - 1|/(|k| + 1) = |10k - 15|/(|10k| + |15|).

Figure 1: Fitting (1, 1) and (10, 15) via y = kx

Its minimization requires the least value of |k - 1|/|k| on the closed interval [1, 3/2] and gives

(k - 1)/k = -(10k - 15)/(10k), k = 1.5^1/2 ≈ 1.22474.

For the first point (2) with less coordinates, the absolute error, relative ones, and unierror are

Δ_{k =? 1} = |k - 1| = 1.5^1/2 - 1 ≈ 0.22474,

δ_{L , k =? 1} = |k - 1|/|k| = (1.5^1/2 - 1)/1.5^1/2 ≈ 0.18350,

δ_{R , k =? 1} = |k - 1|/|1| = (1.5^1/2 - 1)/1 ≈ 0.22474,

δ_{k =? 1} = |k - 1|/(|k| + |1|) = (1.5^1/2 - 1)/(1.5^1/2 + 1) ≈ 0.10102,

and for the second point with relatively greater coordinates those are

Δ_{10k =? 15} = |10k - 15| = 15 - 10 × 1.5^1/2 ≈ 2.75255,

δ_{L , 10k =? 15} = |10k - 15|/|10k| = (15 - 10 × 1.5^1/2)/(10 × 1,5^1/2) ≈ 0.22474,

δ_{R , 10k =? 15} = |10k - 15|/|15| = (15 - 10 × 1.5^1/2)/15 ≈ 0.18350,

δ_{10k =? 15} = |10k - 15|/(|10k| + |15|) = (15 - 10 × 1.5^1/2)/(10 × 1.5^1/2 + 15) ≈ 0.10102.

Or iterate equalizing condition (k⁽ⁿ⁾ nth approximation) transformed in the chosen interval:

(k⁽ⁿ⁺¹⁾ - 1)/(k⁽ⁿ⁾ + 1) = (15 - 10k⁽ⁿ⁺¹⁾)/(10k⁽ⁿ⁾ + 15).

k⁽⁰⁾ = 1 leads to natural results very close to the exact value unlike the least square method:

(k⁽¹⁾ - 1)/(k⁽⁰⁾ + 1) = (3 - 2k⁽¹⁾)/(2k⁽⁰⁾ + 3), k⁽¹⁾ = 11/9 ≈ 1.22222,

(k⁽²⁾ - 1)/(k⁽¹⁾ + 1) = (3 - 2k⁽²⁾)/(2k⁽¹⁾ + 3), k⁽²⁾ = 109/89 ≈ 1.22472.

4.5. Direct solution method

Let us approximate a quantiset A whose elements belong to a common normed vector space [7] and whose quantities are nonnegative real numbers, with using its quantisubset A’ after excluding exactly all the quantielements of A with zero elements

A = {... , _αa , ... , _βb , ... , _γ0 , ...}° ⊇ A’ = {... , _αa , ... , _βb , ... }°

by a quasisolution x (in the same vector space) to the quantiset of equations

... , _α(x = a), ... , _β(x = b), ... , _γ(x = 0), ... .

The essence of the direct solution method is that the element

x = Q(A’)//Q(A)(... + αa/||a||^P + ... + βb/||b||^P + ...)//(... + α/||a||^P + ... + β/||b||^P + ...)

(Q the multiquantity) of the same vector space is at once proposed as a proper approximation to this quasisolution generally unknown where P is a nonnegative real number, we shall take P = 1/2. All zero elements are taken into account not geometrically but arithmetically due to

Q(A’)//Q(A) = ( ... + α + ... + β + ...)//( ... + α + ... + β + ... + γ + ...)

along with all the quantities. For the same overdetermined set of equations (4) we obtain the previous result. For a more complicated overdetermined set of equations such that each of them might include two or more unknown variables, let us consider each of their intersections that simultaneously belongs to t geometric interpretations of the equations to have a quantity t(t - 1)/2 possibly divided by the sum (possibly of the squares) of the distances of the intersection from all the equations interpretations. Possibly consider some quantisubset of the intersections. Then use the method for each variable separately and unite the results for the whole set of the variables. The quasisolution, see Figure 2, to the set of four linear equations

Figure 2: Quasisolution (°) to the set of four equations

x + 2y = 3,

2x - y = 1,

x - y = 1,

2x - 3y = 0:

x ≈ 1.151, y ≈ 0.750; ¹δ ≈ 0.1040, ²δ ≈ 0.1276, ⁴δ ≈ 0.1527.

4.6. Typical examples for comparing different methods

First let us solve the problem in Figure 3 on best fitting the four points

(x, y) = (-1, -1);

(1, 1);

(8, 10);

(12, 10)

of the Cartesian plane x0y by a straight line

y = ax + b

where coefficients a and b are required. Instead of the result graphically evident

y = x ,

the least square method (LSM) badly fitting the first two points with less coordinates gives

y ≈ 0.927 x + 0.364; ¹δ ≈ 0.128, ²δ ≈ 0.135, ⁴δ ≈ 0.146.

Figure 3: Linearly fitting four points (-1, -1); (1, 1); (8, 10); (12, 10) by the least square method and the least normed square method

The least normed square method (LNSM) (0th approximation a⁽⁰⁾ = b⁽⁰⁾ = 1) gives the 1st one

y ≈ 0.998 x + 0.008; ¹δ ≈ 0.053, ²δ ≈ 0.071, ⁴δ ≈ 0.086

already very well corresponding to intuition and obviously bringing much less unierrors.

Now find the quasisolution to the overdetermined set of four linear equations, see Figure 4:

29x + 21y = 50,

50x - 17y = 33,

x + 2y = 7,

2x - 3y = 0.

Figure 4: Quasisolutions to the set of four equations by the least square method, the least normed square method, and the unierror equalizing method

The least square method (LSM) ignoring the last two equations with less factors gives

x ≈ 1.00234, y ≈ 1.00750; ¹δ ≈ 0.151, ²δ ≈ 0.223, ⁴δ ≈ 0.286.

The least normed square method (LNSM) by 0th approximation x⁽⁰⁾ = 1, y⁽⁰⁾ = 1 gives 1st one:

x⁽¹⁾ ≈ 1.263, y⁽¹⁾ ≈ 1.051; ¹δ⁽¹⁾ ≈ 0.163, ²δ⁽¹⁾ ≈ 0.195, ⁴δ⁽¹⁾ ≈ 0.250.

The unierror equalizing method (UEM) brings good further approximations and the result:

x⁽²⁾ ≈ 1.5974, y⁽²⁾ ≈ 1.4374; ¹δ⁽²⁾ ≈ 0.1855, ²δ⁽²⁾ ≈ 0.1879, ⁴δ⁽²⁾ ≈ 0.1925;

x⁽³⁾ ≈ 1.5993, y⁽³⁾ ≈ 1.4674; ¹δ⁽³⁾ ≈ 0.1864, ²δ⁽³⁾ ≈ 0.1884, ⁴δ⁽³⁾ ≈ 0.19219;

x⁽⁴⁾ ≈ 1.5981, y⁽⁴⁾ ≈ 1.4684; ¹δ⁽⁴⁾ ≈ 0.1864, ²δ⁽⁴⁾ ≈ 0.1884, ⁴δ⁽⁴⁾ ≈ 0.19216;

x ≈ 1.60, y ≈ 1.47; ¹δ ≈ 0.186, ²δ ≈ 0.188, ⁴δ ≈ 0,192.

The direct solution method without considering the distances brings the similar quasisolution

x ≈ 1.536, y ≈ 1.432; ¹δ ≈ 0.184, ²δ ≈ 0.187, ⁴δ ≈ 0.192.

The unierror equalizing method and the direct solution method seem to be the most effective.

5. SOME ELASTIC MATHEMATICS APPLICATIONS TO HELICOPTER DESIGN

5.1. Data scatter

All the above concepts and methods of elastic mathematics give best data approximations.

5.2. Movement of the center of gravity

Table 1: Horizontal coordinate x_c.g. of the center of gravity as a function of fuel quantity, piece-linear (index lin) and piece-quadratic (index quadr) approximations of x_c.g., and their absolute and relative deviations from x_c.g.

6. BASIC RESULTS FOR A HELICOPTER UNDER FLIGHT CONDITIONS

Comparing the distributions of moments, shears, and stresses varying by different flight conditions (maximum speed, symmetric and rolling pullout) due to fuel consumption (first separately for the design alternate forward and aft configurations of the helicopter and then unifying all the data to use the envelopes1 created by the both comparison bases) shows:

1. From the nose of the helicopter to its gravity center zone, the stresses holding in the longitudinal vertical plane in the helicopter configurations to be investigated are some more dangerous than the same stress envelope created by the both comparison bases.

2. From the gravity center zone up to the tail, the stresses holding in the longitudinal vertical plane are contained in the same stress envelope created by the both comparison bases.

The obtained results have provided optimizing the helicopter by locally improving the fatigue strength [16] of the relevant elements of the design alternate helicopter configurations.

7. CONCLUSIONS

1. Helicopter design needs optimizing its configurations under flight conditions via adequate analysis of test data scatter and of moving the center of gravity due to fuel consumption.

2. The concepts of the absolute and relative errors and the least square method in classical mathematics have many defects of principle and cannot provide adequately solving these and many other typical problems.

3. Elastic mathematics with its concepts of the unierror, reserve, reliability, and risk along with the unierror and reserve iteration methods of the least normed powers, equalizing, and direct solution is very suitable for solving such typical problems.

4. Applying these concepts and methods of elastic mathematics provides optimizing helicopter configurations under flight conditions and improving its fatigue strength.

8. BIBLIOGRAPHY

1. Bell Reports 205-099-004, 205-099-006, 205-099-007, and 205-099-011.

2. Military Handbook. Metallic Materials and Elements for Aerospace Vehicle Structures. MIL-HDBK-5H, 1998

3. Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998

4. Lev Gelimson: General Strength Theory. Drukar Publishers, Sumy, 1993

5. Lev Gelimson: General strength theory. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 56-62

6. Lev Gelimson: General analytic methods. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 260-261

7. Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

8. Lev Gelimson: Basic New Mathematics. Drukar Publishers, Sumy, 1995

9. Lev Gelimson: Elastic mathematics. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 264-265

10. Lev Gelimson: Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich (Germany), 2004

11. Lev Gelimson: Quantianalysis: Uninumbers, quantioperations, quantisets, and multiquantities. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 15-21

12. Lev Gelimson: Quantisets algebra. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 262-263

13. Lev Gelimson: General estimation theory. Transactions of the Ukraine Glass Institute, 1 (1994), 214-221

14. Lev Gelimson: General problem theory. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 26-32

15. Lev Gelimson: The method of least normalized powers and the method of equalizing errors to solve functional equations. Transactions of the Ukraine Glass Institute, 1 (1994), 209-213

16. Bruhn, E. F.: Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc., Indianapolis (IN), 1973