ИССЛЕДОВАНИЕ ЗАДАЧИ О ФОРМАХ ПРОГИБА СВОБОДНО

ISSN 1810-0198. Âåñòíèê ÒÃÓ, ò. 16, âûï. 4, 2011
e1 = Γ0 a2 (e0 , λ0 ), λ1 = −
ãäå
(a2 (e0 , λ0 ), g0 )
.
A e0 , g0
ËÈÒÅÐÀÒÓÐÀ
1.
2.
Ôèëèïïîâ À.Ô. Äèôôåðåíöèàëüíûå óðàâíåíèÿ ñ ðàçðûâíîé ïðàâîé ÷àñòüþ. Ì.: Íàóêà, 1985.
Þìàãóëîâ Ì.Ã., Èáðàãèìîâà Ë.Ñ. Ôóíêöèîíàëèçàöèÿ ïàðàìåòðà è åå ïðèëîæåíèÿ â çàäà÷å î ëîêàëü-
íûõ áèôóðêàöèÿõ äèíàìè÷åñêèõ ñèñòåì // Àâòîìàòèêà è òåëåìåõàíèêà. Ì., 2007.  4. Ñ. 312.
Ïîñòóïèëà â ðåäàêöèþ 10 àïðåëÿ 2011 ã.
Sharafutdinov I.V. An asymptotic formulae in the problem of bifurcation of stationary solutions in nonsmooth dynamic systems. We consider the problem of bifurcation of stationary solutions
in dynamical systems with nonsmooth right-hand sides. There is proposed a bifurcation criteria
an iterative procedure, and asymptotic formulae for bifurcating solutions.
Key words: bifurcation; nonsmooth system; stationary solution.
Øàðàôóòäèíîâ Èëüäàð Âàêèëüåâè÷, Ñòåðëèòàìàêñêàÿ ãîñóäàðñòâåííàÿ ïåäàãîãè÷åñêàÿ
àêàäåìèÿ, ã. Ñòåðëèòàìàê, Ðîññèéñêàÿ Ôåäåðàöèÿ, êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
ñòàðøèé ïðåïîäàâàòåëü êàôåäðû àëãåáðû, ãåîìåòðèè è ìåòîäèêè îáó÷åíèÿ ìàòåìàòèêå,
e-mail: sh_ildar_79@mail.ru.
ÓÄÊ 517.927
ÈÑÑËÅÄÎÂÀÍÈÅ ÇÀÄÀ×È Î ÔÎÐÌÀÕ ÏÐÎÃÈÁÀ ÑÂÎÁÎÄÍÎ
ÎÏÅÐÒÎÉ ÏËÀÑÒÈÍÛ ÏÐÈ ÏÐÎÄÎËÜÍÎÉ ÍÀÃÐÓÇÊÅ
c Ã. Ã. Øàðàôóòäèíîâà
Êëþ÷åâûå ñëîâà
: êðèòè÷åñêàÿ ñèëà; òî÷êà áèôóðêàöèè; àñèìïòîòè÷åñêèå ôîðìóëû; ñî-
ñòîÿíèå ðàâíîâåñèÿ.
 ðàáîòå ïðåäëàãàåòñÿ ñõåìà ïåðåõîäà îò êðàåâîé çàäà÷è îá èçãèáàõ ïëàñòèí ñî ñâîáîäíî
îïåðòûìè êðàÿìè ïðè ïðîäîëüíîé íàãðóçêå ê îïåðàòîðíîìó óðàâíåíèþ, ïðèâîäÿùåìó
ê àñèìïòîòè÷åñêèì ôîðìóëàì äëÿ ïðèáëèæåííîãî ïîñòðîåíèÿ ðåøåíèé.
Ðàññìàòðèâàåòñÿ çàäà÷à î ïðîãèáå ïðÿìîóãîëüíîé ïëàñòèíû P ïðè äåéñòâèè ïðîäîëüíîé ñæèìàþùåé ñèëû Ny , ïðèëîæåííîé ê êðàÿì ïëàñòèíû âäîëü îñè Oy . Äèôôåðåíöèàëüíûå óðàâíåíèÿ, ñâÿçûâàþùèå ôóíêöèþ Φ íàïðÿæåíèé (ôóíêöèþ Ýéðè) â ñðåäèííîé
ïîâåðõíîñòè è ôóíêöèþ ïðîãèáà W äëÿ ñâîáîäíî îïåðòîé ïëàñòèíû, èìåþò âèä
L1 ≡ d · ∇4 W − h · L(W, Φ) = 0 ,
(1)
1
E · L(W, W ) = 0 ,
2
(2)
L2 ≡ ∇4 Φ +
ãäå ∇4 äâóìåðíûé îïåðàòîð Ëàïëàñà, íåëèíåéíûå îïåðàòîðû
äåëÿþòñÿ ðàâåíñòâîì
L(W, Φ) =
L(W, Φ)
∂2W ∂2Φ ∂2W ∂2W
∂2W ∂2Φ
,
+
−
2
∂x2 ∂y 2
∂y 2 ∂x2
∂x∂y ∂x∂y
è
L(W, W )
îïðå(3)
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ISSN 1810-0198. Âåñòíèê ÒÃÓ, ò. 16, âûï. 4, 2011
èçâåñòíûå ïîëîæèòåëüíûå ïîñòîÿííûå, ïàðàìåòðû ïëàñòèíû.
Äëÿ èññëåäîâàíèÿ çàäà÷è (1)(3) óäîáíî ïåðåéòè ê êðàåâîé çàäà÷å ñ îäíîðîäíûìè ãðàíè÷íûìè óñëîâèÿìè:
h, E, D
1 ≡ d · ∇4 W − h · L(W, F ) + h · Ny · L(W, C) = 0 ,
L
(4)
2 ≡ ∇4 F + 1 E · L(W, W ) = 0 ,
L
2
(5)
2
âåùåñòâåííûé ïàðàìåòð, F (x, y) = Φ(x, y) + Ny · C(x, y), C(x, y) = x2 .
Ïðè ëþáîì çíà÷åíèè ïàðàìåòðà Ny çàäà÷à (4)(5) èìååò òðèâèàëüíîå ðåøåíèå
W (x, y) ≡ 0 , F (x, y) ≡ 0 , îäíàêî íóëåâîå ðåøåíèå íå âñåãäà åäèíñòâåííî. Ýòî ñîîòâåòñòâóåò èçâåñòíîìó ýêñïåðèìåíòàëüíîìó ôàêòó: ïëàñòèíà ìîæåò èìåòü ïðè îäíîé è òîé æå
íàãðóçêå íåñêîëüêî ðàçëè÷íûõ ôîðì ðàâíîâåñèÿ. Êàê ïðàâèëî, ëèøü îäíà èç ôîðì ðàâíîâåñèÿ ÿâëÿåòñÿ æåëàòåëüíîé. Ïåðåõîä â äðóãèå ôîðìû ìîæåò âûçâàòü ðàçðóøåíèå êîíñòðóêöèè. Ïîýòîìó âîçíèêàåò íåîáõîäèìîñòü â ïðåäñêàçàíèè òàêîãî ïåðåõîäà, ÷òî ñâîäèòñÿ
ê îòûñêàíèþ êðèòè÷åñêèõ çíà÷åíèé ñèë Ny , èëè òî÷åê áèôóðêàöèè çàäà÷è (4)(5).
Ñòðîãî ãîâîðÿ, ñ òî÷êè çðåíèÿ îáùåé òåîðèè áèôóðêàöèé, íàëè÷èå êðèòè÷åñêèõ çíà÷åíèé ñèë Ny∗ åùå íå îçíà÷àåò êà÷åñòâåííîãî èçìåíåíèÿ ôîðìû ðàâíîâåñèÿ ïëàñòèíû ïðè
ïåðåõîäå íàãðóçêè ÷åðåç òàêèå êðèòè÷åñêèå çíà÷åíèÿ. Äðóãèìè ñëîâàìè, â çàäà÷àõ î òî÷êàõ
áèôóðêàöèè îáû÷íî ïðèñóòñòâóåò íåîáõîäèìîå è äîñòàòî÷íîå óñëîâèå áèôóðêàöèè. Íåîáõîäèìîå ñâÿçàíî ñ òåì, ÷òî ñîîòâåòñòâóþùèå ëèíåàðèçîâàííûå óðàâíåíèÿ èìåþò íåíóëåâûå
ðåøåíèÿ, à äîñòàòî÷íîå ñâÿçàíî ñ òðàíñâåðñàëüíûì ïîâåäåíèåì ñîîòâåòñòâóþùèõ ñîáñòâåííûõ çíà÷åíèé ëèíåéíîé çàäà÷è. Îäíàêî â çàäà÷å î ïðîãèáàõ ïëàñòèí íåîáõîäèìîå óñëîâèå
îäíîâðåìåííî ÿâëÿåòñÿ è äîñòàòî÷íûì.
Íåíóëåâîå ðåøåíèå ëèíåéíîé êðàåâîé çàäà÷è d · ∇4W = −Ny · hL(W, C) ïðåäñòàâèì
∞ ∞
πmy
sin
. Îòñþäà ëåãêî ïîëó÷èòü êðèòè÷åñêóþ ñèëó, èëè
â âèäå W = Bkm sin πkx
a
b
ãäå
Ny
k=1 m=1
2
2
2 2
òî÷êó áèôóðêàöèè çàäà÷è (4)-(5): Ny∗ = πhd · (a a+4bb2 ) .
Oáîçíà÷èì ÷åðåç B : W2◦2 → W2◦2 è D : W2◦2 → W2◦2 îïåðàòîðû òàêèå, ÷òî
a b
a b
dΔW Δϕdxdy , (DW, ϕ) = −2h
(BW, ϕ) = 2
0
0
L(W, c)ϕdxdy .
0
0
Îïèðàÿñü, äàëåå, íà îáùóþ òåîðèþ áèôóðêàöèé ìàëûõ ðåøåíèé îïåðàòîðíûõ óðàâíåíèé è
íåêîòîðûå ðåçóëüòàòû èç [2] è [3], ïðèâåäåì çàäà÷ó (4)-(5) ê îïåðàòîðíîìó âèäó
W = A(λ)W + a2 (W ) , W ∈ W2◦2 , λ ∈ R ,
ãäå
A(λ) = I + B − λD,
à íåëèíåéíûé îïåðàòîð
a b
a2 (W ), ϕ = −2h
0
L W, A0 (W ) ϕdxdy ,
(B − λD)W = 0
e=
1226
a2 (W ) : W2◦2 → W2◦2
0
Ò å î ð å ì à 1. Óðàâíåíèå
(6)
ïðè
lim
max
W →0 |λ−λ0 |δ
λ = λ∗ = Ny∗
4a3 b3
πx
πy
sin
sin
.
2
2
2
(a + b )
a
b
òàêîé, ÷òî
a2 (W )
=0.
W èìååò íåíóëåâîå ðåøåíèå
ISSN 1810-0198. Âåñòíèê ÒÃÓ, ò. 16, âûï. 4, 2011
2 7 5
8hπ a b
Òåîðåìà 1 äàåò íåîáõîäèìîå óñëîâèå áèôóðêàöèè. Óñëîâèå −(De, e) = − (a
= 0
2 + b2 )5
îáåñïå÷èâàåò âûïîëíåíèå äîñòàòî÷íîãî óñëîâèÿ áèôóðêàöèè.
Ò å î ð å ì à 2. Çíà÷åíèå λ∗ = Ny∗ ÿâëÿåòñÿ òî÷êîé áèôóðêàöèè çàäà÷è (4)(5).
Ò å î ð å ì à 3. Áèôóðöèðóþùèå ðåøåíèÿ Wε óðàâíåíèÿ (6) è ñîîòâåòñòâóþùèå çíà-
÷åíèÿ ïàðàìåòðà λε = λ(Wε ) ïðåäñòàâèìû â âèäå
Wε = εe + ε2 e1 + o(ε2 ) , λε = λ∗ + ελ1 + o(ε) , ε 0 ,
ãäå λ1 è e1 ìîãóò áûòü âûïèñàíû â ñîîòâåòñòâèè ñ [3].
ËÈÒÅÐÀÒÓÐÀ
1.
2.
Òèìîøåíêî Ñ.Ï., Âîéíîâñêèé-Êðèãåð Ñ. Ïëàñòèíêè è îáîëî÷êè. Ì.: Íàóêà, 1966.
Áîáûëåâ Í.À., Åìåëüÿíîâ Ñ.Â., Êîðîâèí Ñ.Ê. Ãåîìåòðè÷åñêèå ìåòîäû â âàðèàöèîííûõ çàäà÷àõ. Ì.:
Ìàãèñòð, 1998.
3.
Þìàãóëîâ Ì.Ã., Èáðàãèìîâà Ë.Ñ. Ôóíêöèîíàëèçàöèÿ ïàðàìåòðà è åå ïðèëîæåíèÿ â çàäà÷å î ëîêàëü-
íûõ áèôóðêàöèÿõ äèíàìè÷åñêèõ ñèñòåì // Àâòîìàòèêà è òåëåìåõàíèêà. Ì., 2007.  4. Ñ. 312.
Ïîñòóïèëà â ðåäàêöèþ 10 àïðåëÿ 2011 ã.
Sharafutdinova G.G. Study of the problem of depression forms of freely supported plate under
for longitudinal force. In the paper there is proposed a scheme of transition from the boundary
value problem for bending of plates with freely supported edges under longitudinal force to
the operator equation, that leads to asymptotic formulae for the approximate construction of
solutions.
Key words: critical force; bifurcation point; asymptotic formulae; balance state.
Øàðàôóòäèíîâà Ãþçåëü Ãàôóðîâíà, Ñòåðëèòàìàêñêàÿ ãîñóäàðñòâåííàÿ ïåäàãîãè÷åñêàÿ
àêàäåìèÿ, ã. Ñòåðëèòàìàê, Ðîññèéñêàÿ Ôåäåðàöèÿ, ñòàðøèé ïðåïîäàâàòåëü êàôåäðû ìàòåìàòè÷åñêîãî àíàëèçà, e-mail: guzelbas@mail.ru.
ÓÄÊ 517.51
ÎÁ ÎÃÐÀÍÈ×ÅÍÍÎÑÒÈ ÌÍÎÆÅÑÒ  ÏÐÎÑÒÐÀÍÑÒÂÀÕ
ÌÓÑÅËßÊÀ-ÎÐËÈ×À
c È.Â. Øðàãèí
Êëþ÷åâûå ñëîâà
: îãðàíè÷åííîå ìíîæåñòâî; êâàçèíîðìà; ïðàâèëüíîå ïðîñòðàíñòâî; ãåí-
ôóíêöèÿ; ïðîñòðàíñòâî ÌóñåëÿêàÎðëè÷à.
 êâàçèíîðìèðîâàííûõ ïðîñòðàíñòâàõ ÌóñåëÿêàÎðëè÷à ðàññìàòðèâàþòñÿ ïîíÿòèÿ
ìåòðè÷åñêîé è òîïîëîãè÷åñêè-âåêòîðíîé îãðàíè÷åííîñòè ìíîæåñòâ. Ïðîñòðàíñòâî íàçûâàåòñÿ ïðàâèëüíûì, åñëè äëÿ íåãî îáà îïðåäåëåíèÿ ýêâèâàëåíòíû. Óñòàíîâëåíî íåîáõîäèìîå óñëîâèå ïðàâèëüíîñòè ïðîñòðàíñòâà è óêàçàíû íåêîòîðûå äîñòàòî÷íûå óñëîâèÿ.
Êàê èçâåñòíî, îãðàíè÷åííîñòü ìíîæåñòâà îïðåäåëÿåòñÿ â ìåòðè÷åñêèõ è â òîïîëîãè÷åñêèõ âåêòîðíûõ ïðîñòðàíñòâàõ ïî-ðàçíîìó. Ïðè ýòîì äëÿ íîðìèðîâàííûõ ïðîñòðàíñòâ îáà
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