7_Romanuk.doc Assertion on off-bound components in nonregular projector optimal strategy for a construction with four supports within three identical … Проблеми трибології (Problems of Tribology) 2011, № 3 36 Romanuke V.V. Khmelnytskyy National University, Khmelnytskyy, Ukraine ASSERTION ON OFF-BOUND COMPONENTS IN NONREGULAR PROJECTOR OPTIMAL STRATEGY FOR A CONSTRUCTION WITH FOUR SUPPORTS WITHIN THREE IDENTICAL PARTIAL UNCERTAINTIES OF COMPRESSIONS Actuality and essentiality of the problem There is a known problem of distributing optimally building resources in the support construction, where each support is under partially uncertain pressure [1]. It is ever hard to before-evaluate those partial uncer- tainties (PU), and so they often are laid to be identical, what, on the one hand, simplifies the model of distribu- tion, but, on another, necessitates asserting some claims to resolve the problem faster. Analysis of recent investigations on removing partial uncertainties in constructing supports Construction with four supports is a classic building support, where their axes are in nodes of rectangu- lar. If the total compression on the construction is unit-normed (UN), than even within PU of compressions on each support there is a relevant model, guaranteeing the full reliability in final result. This model is an antagonis- tic game (AG), which kernel [1] for the classic support construction ( ) ( ) ( ) ( ) ( ) ( ){ }1 2 3 1 2 3 1 1 1 2 2 2 3 3 3 4 4 4, , , ; , , max , , , , , , ,T T x x x y y y T x y T x y T x y T x y= = =X Y ( ) ( ) 1 2 3 1 2 3 1 2 3 1 2 3 2 22 2 2 2 2 2 1 2 3 1 2 31 2 3 1 2 3 1 1 max , , , max , , , 1 1 x x x x x x x x x x x x y y y y y yy y y y y y    − − − − − −    = α α α α = α    − − − − − −       (1) is defined on the Cartesian product (2) of two parallelepipeds (3) and (4) as the sets of pure strategies (5) and (6) of the first and second players, respectively. The value ix is the UN pressure on the i -th support, being selected by the first player, and the value iy is the UN cross-section square (CSS) of the i -th support, being selected by the second player, might be called the projector for further, 1, 3i = . The condition 1, 4s∀ = (7) means that every support is compressed with nonzero force, whence there is a condition 3 1 1i i b = <∑ . (8) Other conditions, s sa b< 1, 4s∀ = and 3 1 1i i a = <∑ , are consequent from (7) and (8) correspondingly. And, whatever, 4 4 1 1 1s s s s x y = = = =∑ ∑ . PDF created with pdfFactory Pro trial version www.pdffactory.com http://www.pdffactory.com http://www.pdffactory.com Assertion on off-bound components in nonregular projector optimal strategy for a construction with four supports within three identical … Проблеми трибології (Problems of Tribology) 2011, № 3 37 Purpose of the paper Assume, that the before-evaluated partial [ ]{ }3 1 ;i i ia b = -uncertainties had appeared to be identical: ib b= and ia a= 1, 3i∀ = . The matter is what the components of projector optimal strategy (POS) (9) will be, if POS appears nonregular [1]. The purpose of the current paper is to make some assertions on this. Recollecting the convexity of AG with kernel (1) on hyperparallelepiped (2) First above all, recall that AG with kernel (1) on hyperparallelepiped (2) is convex [1]: 1, 3i∀ = (10) as almost everywhere [1] (but minding the zero derivatives) 2 2 2 3 4 2 6 0i i i i i i i i x x x y y y y y    ∂ ∂ = − = >    ∂ ∂    1, 3i∀ = , (11) ( ) ( ) ( ) ( ) ( ) 2 1 2 3 1 2 31 2 3 2 3 42 1 2 3 1 2 3 1 2 3 2 1 6 11 0 1 1 1i i x x x x x xx x x y yy y y y y y y y y    − − − − − −∂ − − − ∂    = = >    ∂ ∂− − − − − − − − −    1, 3i∀ = . (12) Surely, (10) is also true everywhere [1]. And this convexity gives the single POS (9) with components * 3 3 1 2 3 1 2 3 1 1 1 1 i i i k k k k b b y b b b a a a b a = = = = + + + − − − + −∑ ∑ , 1, 3i = . (13) However, (13) is true only if [ ] 3 3 1 1 ; 1 i i i k k k k b a b b a = = ∈ + −∑ ∑ 1, 3i∀ = . (14) Especially this agitates the interest when compressions had been before-evaluated as three identical [ ];a b - uncertainties, and it matters badly if then the component * 1 2 3 1 2 31 3 1 3 i i b b y b b b a a a b a = = + + + − − − + − (15) is out of the range, that is POS appears to be off-bound (with all its components simultaneously). Theorems on left and right off-bound (nonregular) POS Theorem 1. In AG with kernel (1) on (2) by three identical [ ];a b -uncertainties and there is POS [ ]* a a a=Y . (16) Proof. Having 3 1 3 b a b a < + − (17) causes impossibility of the statement [1] ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 1 2 3 * 2 2 2 2 2 2* * * * * * * * * * 1 2 3 1 2 3 1 2 3 1 1 3 1 1i b b b b a a a a v y y y y y y y y y y − − − − = α = α = α = α = α = α − − − − − − (18) with components (15) for the optimal game value *v . However, by * iy a= 1, 3i∀ = gives PDF created with pdfFactory Pro trial version www.pdffactory.com http://www.pdffactory.com http://www.pdffactory.com Assertion on off-bound components in nonregular projector optimal strategy for a construction with four supports within three identical … Проблеми трибології (Problems of Tribology) 2011, № 3 38 ( ) 1* 1 3v a − = α − and (16) as ( ) ( )2 2 2* * * *1 2 3 1 3 1 1 31i b b a a ay y y y − α = α < α = α −− − − under maximum in (1). Mind that here 1 3 a b< < due to 43 1b b= − and 4 0b > . So, 3 1 3b a b a a< + − , ( )1 3 1 3b a a a− < − , ( )1 3b a a− < , ( ) 21 3b a a− < , 2 1 3 a b a < − , , , , . (19) The roots of the corresponding equation 23 3 1 0a a+ − = are 1 3 21 6 a − − = and 2 3 21 6 a − + = . As 1 2 10 3 a a< < < , then (17) is true by , causing the left off-bound (nonregular) POS (16). The theo- rem has been proved. Theorem 2. In AG with kernel (1) on (2) by three identical [ ];a b -uncertainties and there is POS [ ]* b b b=Y . (20) Proof. Having 3 1 3 b b b a > + − (21) causes impossibility of the statement (18) with components (15) for the optimal game value *v . However, by * iy b= 1, 3i∀ = gives 1 *v b −= α and (20) as ( ) ( ) ( )2 2 2* * * *1 2 3 1 1 3 1 3 1 31i b a a b by y y y − − α = α > α = α −− − − under maximum in (1). So, 1 3 1 3 b b a > + − , ( )1 3 1 3b b a> + − , ( ) ( )21 3 1 3b b a− > − , 21 7 9 3b b ab− + > − , 23 9 7 1ab b b> − + − , , . (22) The roots of the corresponding equation 29 7 1 0b b− + = are 1 7 13 18 b − = and 2 7 13 18 b + = . As 1 210 1 3 b b< < < < , then (21) is true by , causing the right off-bound (nonregular) POS (20). The theorem has been proved. Conclusion Conditions and for proved POS (16) and (20) may be restated as 21 3 1 ; 6 3 a  − ∈     or 21 3 ; 6 a b  − ∈     and 7 13 0; 18 b  − ∈     or 7 13 ; 18 b a  − ∈     . Those ones ought to be applied for fast determining the off-bound optimal CSS within three identical PU of compressions. References 1. Романюк В. В. Регулярна оптимальна стратегія проектувальника у моделі дії нормованого одиничного навантаження на N -колонну будівельну конструкцію-опору / В. В. Романюк // Проблеми трибології. – 2011. – № 2. – С. 111-114. PDF created with pdfFactory Pro trial version www.pdffactory.com http://www.pdffactory.com http://www.pdffactory.com