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1MEXANIKA
UDK: 539.3
FAZOVIY YUKLANISHLARDAGI UCH QATLAMLI STERJENLARNING KUCHLANGANLIK-DEFORMATSIYALANGANLIK HOLATLARINI MATEMATIIK MODELLASHTIRISH
Shokirov Davron Abdugaffor ogli Namangan muhandislik-qurilish intituti, shokirov1004@gmail. com 160103, O'zbekiston, Namangan, I. Karimov ko'chasi, 12.
Annotatsiya. Maqolada fazoviy yuklanishlardagi uch qatlamli sterjenning kuchlanganlik-deformatsiyalanganlik holatini tadqiq etishning matematik ta'minotini keltirilgan bo'lib, ushbu ta'minotni ishlab chiqishda Ostrogradskiy-Gamilton variatsion tamoyili asosida matematik model yaratilgan. Uch qatlamli sterjenlarning tebranish tenglamalari mos ravishda umumlashgan boshlang'ich va tabiiy chegaraviy shartlari bilan ishlab chiqilgan. Qo'yilayotgan masalaning hisoblash algoritmini markaziy chekli ayirmalar usulida ishlab chiqilgan. Masalani yechishda ushbu usulning oshkormas sxemasidan foydalanilgan. Hisoblash algoritmida ikkinchi tartibli differensial tenglamalarni matrisali haydash usulidan foydalanilgan natijalar olinishi ko'rsatib o'tilgan.
Annotation. In this article presents a mathematical interpretation of the study of the stress-deformation state of three layered rods under spatial loads and in this development, a mathematical model was created based on the Ostrogradskiy Hamilton variation principle. The calculation algorithm of the given problem was developed by the method of central finite differences. The vibration equations of three-layered rods are derived with appropriate generalized initial and boundary conditions. An undisclosed scheme of this method was used to solve the problem. In the calculation algorithm, it is shown that the results are obtained using the method of matrix driving of second- order differential equations.
Аннотация. В данной статье представлена математическая интерпретация исследования напряженно-деформационного состояния трехслойных стержней при пространственных нагрузках и в этой разработке создана математическая модель на основе вариационного принципа Гамильтона Остроградского. Оно соответствует уравнениям вибрации трехслойных стержней, естественным начальным и граничным условиям. Для решения этой задачи был использован метод центральных конечных разностей и схема неразглашения. В алгоритме расчета показано, что результаты получены методом матричного управления дифференциальными уравнениями второго порядка.
Kalit so'zlar: tebranish, uch qatlamli sterjen, matematik model, Guk qonuni, kuchlanganlik, deformatsiyalanganlik, ko'ndalang egilish, ko'chish, Ostrogradskiy-Gamilton variatsion tamoyili, chekli ayrmalar, haydash usuli, kinetik energiya, potensial energiya, tashqi kuchlar.
Ключевые слова: вибрация, трехслойная стержень, математическая модель, закон Гука, напряжение, деформация, поперечный изгиб, смещение, вариационный принцип Остроградского-Гамильтона, конечные разности, способ вождения, кинетическая энергия, потенциальная энергия, внешние силы
Keywords: vibration, three-layer rods, mathematical model, Hooke's law, stress, deformation, transverse bending, displacement, Ostrogradskiy-Hamilton variational principle, finite differences, driving method, kinetic energy, potential energy, external forces.
Respublikamizda ko'p qatlamli sterjen turiga oid tuzilmalarni nazariy asoslarini takomillashtirish va hisoblash usullarini ishlab chiqish bo'yicha ko'plab olimlar o'z ilmiy
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MEXANIKA
izlanishlarini olib borishgan, jumladan, akademik V.Q.Qobulov tomonidan kostruksion elementlarining chiziqli deformatsiyalanish jarayonlarini aniqlashtirilgan nazariyasi ishlab chiqilgan va amaliy masalalarni yechishga algoritmik yondashuvlar taklif etilgan [1]. O'zbekistonda ko'plab turdagi mexanika masalalarini algoritmlash va avtomatlashtirish uchun algoritmik tizimlar yaratish dastlab akademik V.Q.Qobulov tomonidan taklif etilgan hamda algoritmlash nazariyasi ishlab chiqilgan, akademik T.Bo'riyev, K.Sh.Bobomurodov, F.B.Badalov, B.Kurmanbayev, T.Yuldashev, Sh.A.Nazirov, X.Eshmatov, B.Mardonov, M. Usarov, B.Babajanov va ularning shogirdlari tomonidan rivojlantirilgan.
Masalani qo'yilishi. Bo'ylama, ko'ndalang kuchlarning birgalikdagi ta'sirida fazoviy yuklanishlardagi uch qatlamli sterjenlar nuqtalarining ko'chish masalalarini matematik modellarini Ostrogradskiy-Gamilton variatsion tamoyili asosida ishlab chiqiamiz. Uch qatlamli sterjenning yuk ko'taruvchi qatlamlari uchun Bernulli gipotezasi o'rinli, to'ldiruvchi qatlam balandligi bo'ylab nuqtalar ko'chishining chiziqli approksimatsiyasi bilan elastiklik nazariyasining aniq munosabatlari o'rinlidir.
Mazkur masalaning matemtik modelini ishlab chiqish Ostrogradskiy-Gamilton variatsion tamoyilini quyidagicha ko'rinishida olamiz [1]:
sJ{K-n + A)dt = 0. (1)
t
Bu yerda, (K-n + A) kinetik va potensial energiyalar, tashqi kuchlarning bajargan ishi. Uch qatlamli sterjenda nuqtalarining ko'chishi quyidagicha bo'ladi [2, 3]:
—
u(1) = u, -| z - c —1 la
—
w(1) = w (c < z < c + —);
ul2) = u2z + c +2 Ja ', w(2) = w2(-c-h2 < z <-c); u (3) =fl + z If I u + — a« Vi 1 - z If I u - a(2) 1,
(2)
w(3) = I fl + - J w - - j w2, (-c < z < c ).
Bu yerda bo'ylama va ko'ndalang ko'chishlar u(k)(x,z) va w(k)(x,z), k-qatlam raqami,
—, —, — = 2c - 1, 2 va 3 qatlamlarning qalinliklari.
Asosiy qism. Uch qatlamli sterjenning kinetik energiya variatsiyani aniqlash. Koshi mnosabatlariga ko'ra va (2) formulaga asosan fazoviy yuklanishlardagi deformatsiya komponentalari quyidagi ko'rinishda aniqlanadi [4]:
s = —
•j 2
du duj duk duk
dx. dx dx. dx
V J • • J J
(3)
Kinetik energiyaning variatsiyasini hisoblashda quyidagi munosabatdan foydalanamiz:
JöKdt = \\p^ ^^u^dvdt + d^Suf)dvdt + |Jp]T ^llsufctvdt. (4)
a t
d t
t t v k=1 d t v k=l d t v k=1
Bu erda p - materialning zichligi, bir necha matematik hisoblashlar (differensallash, variatsiyalash, integrallashlardan) so'ng quyidagiga ega bo'lamiz:
(2)-formulani kinetik energiyaning variatsiyasini hisoblash (4) ifodasiga qo'yib, keltirib chiqarilgan formuladan ushbu belgilanishlar hosil bo'ladi.
pF = JJpdzdy, pSy = JJpzdzdy, ply = JJ pz2 dzdy,
y z y z y z
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<
1
MEXANIKA
F - sterjen ko'ndalanag kesim yuzi, Sy - statik moment, Iy - inersiya momenti.
Guk qonuniga asosan kuchlanganlik-deformatsiyalanganlik holatlarini bog'liqligi quyidagicha bo'ladi [5]:
(k) ( ßu(k) du(k) ^
-(.k) _ J7 e(k) _ F dui . _ _ n ßu1 ^ ßu3 °11 - Ekb11 - E
ßx
a - G?
ßz ßx
(5)
Bu yerda E- elastiklik moduli, G^-siljish moduli, a -kuchlanish.
jn-]LT jja^dtdv - ||(an(1) &rn(1) + an(2ön(2) +
(2)
i-1 J-1 v t v t (6)
(3) (1)&13(1) (2) +a3 (3)&r13 (3)) dtdv;
(5) dan kelib chiqib potensial energiyalarni hisoblashda (2) ni (6) ga qo'yib quyidagi tenglikga ega bo'lamiz:
jöMt -jji<)
f ßu(1)
ßx
-öz
ßa
(1)
ßa
(1)
h ßa
(1)
+ öc-+ ö-
ßx ßx 2 ßx
(2)
.ßu(2)
h
A
-öza(2>-öca(2)-ö ^ a(2) ßx 2
1 (3) ( _ ßu(1) _ h ßa(1) _ z ßu(1) Qzh ßa ~(3) ö-+ aö ' *
(1)
2 11 ^ ßx
2 ßx c ßx c 2 ßx
„ ßu(2) h ßa(2) c z ßu(2) z h ßa(2) ^ ö--ö —--ö--+ ö—-- +
(7)
ßx
2 ßx c ßx c 2 ßx
1 (3) ( c- ßW e z ßw, e c- z ßw
+ -a1(3)| ö—- + ö—— + ö —--ö—~ +
2
ßx c ßx ßx c ßx
+ö1 u(1) +öh-a(1) -ö1 u(2) +ö^a(2) I \dvdt. c 2c c 2c JJ
(7) da bir necha matematik hisoblashlar (differensallash, variatsiyalash, integrallash,
belgilashlar) dan so'ng quyidagiga ega bo'lamiz:
Bu yerda N- normal kuchlanganlik, M- kuchlanganlik momenti, Q- urinma kuchlanish.
N(1 - j j a(1)dzdy; M® - j j aumz dzdy; n^ -j j a /2)dzdy; m1(12) -j j a™z dzdy;
y z y z y z y z
n1(13) - j ja„(3)dydz; M^ - j ja„(3)zdydz; Q3 dzdy; - jjza%dzdy.
y z y z y z
Tashqi kuchlar bajargan ishini quyidagi ifoda orqali aniqlaymiz:
y z
jöAtdt -jj
T F(k )öul (k )dvdt + T F3 (k )öw3 (k )dvdt
k-1
jj
k-1 3
+
jj
t l
T q(k )öul (k )d^dt + T q(k )öw3 (k )d^dt
k-1
k-1
+
(8)
T f(kö)d^dt + T f(k öu()dsidt
k-1 k-1
Bu yerda P-hajmiy kuchlar, g-yuzaga ta'sir etuvchi kuchlar, f-chetki kuchlar.
Tashqi kuchlar bajargan ishini hisoblaymiz. Buning uchun (2) gaH uv u larni (8) ga qo'yib, quyidagilarga ega bo'lamiz:
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t v
t v
\SAdt =JJ
MEXANIKA
F( 1 ô Iu( 1 ) - (z - c -h)a( 1 )J + F(2)ô[u(2) - (z + c + ^)a(2)J +
+F
(3)ôI(1 + z)(!u( 1 ) + ha(1 )) + (1 - z)(iu<2) - h2-a(2))J +
c 2
c 2
+F° }ÔWi + F^ôw, + F^ôi1 (1 + -)wi + 1 (1 - -)w2
I 2 c 2 c
+JJ
t i
dvdt +
h
q(1)ô I u(1) - (z - c - |)a(1) J + q(2)ô I u(2) - (z + c + ^)a(2) I +
,(3ô
(1);
Zv L. (1^ h-
+q(3)ôI (1 + zX1u(1) + ha(1)) + (1 - u(2) - ^a(2)) | + 1 1 c 2 4 c 2 4 J
+ q^ôw, + q®ôw2 + q3(3)ô [ 1 (1 + Z K +1 (1 - Z K
I 2 c 2 c
(3)
didt +
(9)
1"ô
+ 11 t, ôu(1) -Mf[(r>ôa(r> + cf;'ôa(l) + h+ 77ôu(2) -^^^ô2) -
;(1),
h 7(1);
;(2) ,
2
2'
4
-c7(2)ôa(2) -hJ^ +1]/(3)ôu(1) + ¡3)ôa(1) + -1 M7((3ôu« + 1 1 1 1 2c 1
+hLMtj(3)ôa(1) +17j(3)ôu(2) - h27(3)ôc(2) - Mf^ôu(2) + ^M7((3)ôa(2) 4c 2 4 2c 4c
;(1 ) ^ —(2^ 1 -(3^ 1 , 1 -(3^ 1
dt
+f 3 ôW1 + 73 ôw2 + - 73 ôW1 + —Mf( )ôW1 +- 73 ôw2 - —Mf3( ^
2 2c 2 2c
Qavslarni ochib yuborib, sterjenga ta'sir etuvchi hajmiy kuchlar ifodasini y va z o'qlar bo'yicha integrallash, yuzaga ta'sir etuvchi kuchlarni esa x o'qi bo'yicha (sterjenning uzunligi) bo'yicha integrallash amallarini bajargandan so'ng quyidagilarga ega bo'lamiz:
J Adt = JJ
J J F {l)ôu{l)dydz - J J F(1) zôamdydz + c J J F {l)ôamdydz + h J J F {l)ôamdydz +
+J J F(2)ôu{2)dydz - J J F(2)zôa(2)dydz - -c J J F{2)ôa(2)dydz - ^ J J F{2)ôa(2)dydz +
y z
y z
y z
y z
+ -2
+ -2
+
1JJ F (3)ôu ()dydz + ^ JJ F (3)ôa(1) dydz + — JJ F(3) zôu {1)dydz + h\J F?3) zôa( l)dydz + 2 4 2c 4c
y z y z y z y z
1 J J F (3)ôu (2)dydz - ^ J J F (3)ôa(2) dydz - — J J F(3) zôu (2)dydz + h f fF (3) zôa(ï)dydz +
2 J J 4JJ 2c ^ 4c ^
y z y z y z y z
J J F{l)ôwldydz + J J F(2)ôw2dydz +1J J F3(3)ôwldydz + — J J F3(3)zôw.dydz +
? z y z y z
1J J Fsôw2dydz - — J J F(3) zôw2dydz
+ — 2
dxdt +
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t v
t x
y z
y z
yz
yz
MEXANIKA
+J J q(lSu (l)dl -J q(1)zSa(V>dl + c J q(1)Sa(1)di + K J q(1)Sa(1)di + J q(2)Su (2)di -t _ i i i 2 i i
-J q(2)zSa(2)di - c J q{2)Sa(2)di - ^ J q(2)Sa(2)di + - J q(3Su(1)di + ^ J q(3) Ja(1)di + i i 4 i 2 i 4 i
+ — fq/3)zSu(1)di + A. fq^zadi + - fq/3)Su(2)di -^ fq(3)Ja(2)di - — f q^zSu(2)di + (10) 2c i 4c , 2 , 4 , 2c ,
+ A J q(3) zSa<2) di + J q (l)Swldi + J q(2) Sw2di + - J q {3)Swldi + ^ J q3 (3) zSw^di + +—J q3^3Sw2di --— J q3^zSw2di dt + J
J f (l)Su(l)ds, - J f(1)zöa^ds, + cJf (l)Sa{X)ds, +
+ — 2
| Jf (1)Sa(1)dsx +Jf (2)Su(2)ds - Jf(2)zöa(2)dsl - cJ f {2)öa(2)ds, - hJf 2)Ja(2)ds1 +
— Jf (3)Ju(1)ds + ^ Jf (3)Ja(1)ds + — Jf(3)zJu(1)ds + A Jf(3)zSa^ds, + 1Jf (3Su^
I sl sl sl sl
Jf(3W2)d^, - — Jf(3)zSu(2)ds + — J f(3) zSa(2) ds + J f ^1)Jw1ds1 + J f(2^Sw2d^ + s1 s1 s1 s1
+ — J tfSwfa + ^ J f3(3) zSw—ds— + — J f^'Sw2dsl-A- J f3(3) zSw2dSj
+ — 2
h 4
Bu erda
Fi = JJFmdydz, Fi = JJF(2)dydz, Fi = JJF(3)dydz, MF,(1) = JJF(1)zdydz,
y z yz yz y z
MF(2) = JJF(2)zdydz, MF(3) = JJF(3)zdydz, F? = JJF3(1)dydz, F32) = JJF3(2)dydz, (11)
y z y z y z y z
F 33) = JJf3(3) dydz, MF3(1) = JJ F3(1) zdydz, MF3(2) = JJ F3(2) zdydz, MF3(3) = J J F3(3) zdydz;
q(1) = J J q(1)dydz, q(2) = J J q^dydz, q(3) = J J qf)dydz, Mqf ) = J J q(1)zdydz,
yz yz y z y z
Mq(2) =JJq(2)zdydz, MqJ3 =JJq(3)zdydz, q(1) =JJq^dydz, q(2) =JJq(2)dydz,
yz yz yz yz
q(3) = JJ qfdydz, Mq(1 = J J q<zdydz, Mq<2) = J J q(2) zdydz, Mq<3) = J J q<3) zdydz.
Kinetik energiya (4), potensial energiya (7) va tashqi kuchlar bajargan ish (9) lardan ishlab chiqilgan ifodalarni (1) ifodaga qo'yib, fazoviy yuklanishlardagi uch qatlamli sterjenlarning tebranishlari umumlashgan tenglamalar sistemasi va tabiiy boshlang'ich va chegaraviy shartlarini hosil qilamiz [6-10].
Fazoviy yuklanishlarda uch qatlamli sterjenni umumlashgan tenglamalari:
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s
s
s
s
X
yz
yz
yz
yz
yz
yz
yz
2„(1)
pF
d u
MEXANIKA 2^(1) u i f
V
dt2
+ pcF
d2 a
- + p — F
— „ d a(
dt2 2 dt2
1
- + — 4
d2u(1) —
pF
^(i)
+ — pF
d2a
V
dt2 2 dt2
1 , d2u(1) — T d2a(1) ^d2u(2) — ^d2a(2) 1 d2u(2)
+ —^plv-7- + 9 plv-r" + pF-Ö---2 pF-;---TpF-+
c2 y dt2 2c2 y dt2 dt2 2 dt2 c2 y dt2
+
—
2c
2 ply
d 2a(2) 11 fd#1(1) 1 dN? 1 dM,(3) 1 ^ ,1
a
JJ
11
-+--+--^- — Q
dx 2 dx 2c dx 2c
(3) 13
f— (1) 1~(3) 1,^(3) 1 (-(1) 1"(3) 1,^(3)
+1F( + 2 F( + 2cMF1 M «1 + 2 « + 2cMq1
J
Su(1) = 0
pF
2„ (2)
d2u
dt2
- pcF
2„ (2)
d2a
— „d2a(2) 1 f ^d2u(1) — ^d2a(1)
-p-^ F
dt2 2 dt2
- + — 4
pF
- + -1 pF-
dt2 2 dt2
1 , d2u(1) k T d2a(1) ^d2u(2) — ^d2a(2) 1 d2u(2)
-—^plv-,---V ply-+ pF-ö---2 pF-+ —rplv-^
c2 y dt2 2c y dt2 dt2 2 dt2 c2 y dt2
—
2 (2)
2c
ply
d2a
2 r y
dt2
f dN,(,2) 1 dN(3 ) 1 dM,(,3 ) 1 „(3 ) 1
dx 2 dx 2c dx
+— Q
2c 1
+VF " + 2 F ' - -¿MFr J+V + 2 «(3) - ¿M«,m
Su(2) = 0
( ^d2u(1) d2u(1) rd2a(1)
pcF-+ p — F-— + ply-
dt2 2 dt2 y dt2
_ d 2a(1) ^ d 2a(1) + pc F^^ 1 — 1
— d 2a(1) —2 d2a
2
+p — cF 2 dt2
+ F-4 dt2
1
■ + — 4
dt2
— d 2u(1) —
- + pc—F-
2 dt2
- pF- , 2 dt2
d2a
+ pF-4 dt2
+
—1
2«fl)
2c2
pl
du
2 ^ y
dt2
+
— r d2a(1) — d2u(2) —— ^d2a(2) — r d2u(2) —— r + — ply^T + ~ ---— pF plv^T~ + TT ply
d 2a(2)^
+
4c
f
V
dt2
dt2
dt2
2c
2 ^ y ^2
dt2
4c
2y
dt2
JJ
dN11)+— —_ d_NL+—L ML — dQH1+(^A^d^ +
dx dx 2 dx 4 dx 4c dx 4c dx v
—1 —(1) fy — (3) — (3) 1 f (1) -(1) fy-a) — -(3) —
+—F1 + —F1 +— MFC) i +1 -Mq\ ' + cq} +— « + —« + —Mq(3) 2 4 4c y v 2 4 4c
Sa(1) = 0
' rd2u(2) rd2u(2) rd2a(2) 2l7
— ^ d a
(2)
da(2)
dt2 p 2 F dt2 ' py dt2 ' p F dt2 ' 2 F dt2
■ + pc—F
— ^ d 2a( 2) —2 d 2a(2) 1 f — ^ d 2u(1) —— ^ d 2a(1)
^--i-n—^F--1—--L nF---L^LpF-
+pcF- 9 2 dt2
■ + F - , 4 dt2
■ +---2 pF- ,
4, ^ dt2
V
4
a
d?
+
+
—
d2u(1) —— d2a
2c2
pl
^ y ^2
dt2
4c
2 p ly
dt2
- — pF
d 2u(2) „d 2a(2)
dt2
^^^ pF-4
dt2
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MEXANIKA
a
2c:
Ply
d 2u
dt2
A
4c:
2^(2A 1
d2 a
dt2
f
JJ
dM1(12) d<2) h dN1(12) — c
dx
dx
2 dx
h dN^ h2 dM(3 h2 dÖ!(33) 1
4 dx 4c dx 4c dx
+f-MF(2) - cF12) - h F 12) - h2F13) +
I 1 2 4
(12)
+ ^MF,'3 \(-Mq(2)3a(2) - cq? - A qf - A q? + ^Mq(3)
Sa( 2) = 0
PF
d2 w 1
dt2 4
PF
d 2w
1 d2 w1
, d 2w,
dt
1 + p~7 !v—^ +pF—-r -p—; !v
c2 v dt2 -2 v
dt
d_V
dt2
+
1 dQ
(3) 13
1 dM,(3) 1 (-(1) 1 —(3) 1
2 dx 2c dx
-(1) 1 (3)
+1 F31) +1F33) + AMF^ | +1 q(1) +1 q( + AMq,\3)
2c
2c
Swl = 0
( ^ d2w 1 ( ^ d2w 1 . d2w-
pF
dt2
- + -4
d V
PF^T-P—h^r + PF^T + P—1
1 . d2 w.
2
dt2
dt
dt2
,2 y
dt2
+
1 dQ(3)
13
1 dM1(33) 1
2 dx 2c dx
T;(2) 1 T;(3)
+ 1 F 3 +- F 3--MFF
V 2 2c
(3)
f
Tabiiy umumlashgan boshlang'ich shartlar
du(1) ^da(1)1 h ^ da(1) 1
pF--+ pcF -
dt
dt
■ + p — F-
2
dt
+ —
4
pF
du
+
(1)
-(2) 1 (3) 1 , , q3 +- q3 Mq3
(3) 1
Sw, = 0
h da
+ — pF dt 2
(1) h T da(1)
+—V piv—+
dt 2c2 v dt
du(1) dt
du(2) K ^da(2) 1 T du(2) h
+pF---1 pF
dt
dt
da( 2)
dt 2c
da(2) dt
Su(1)
= 0
ßu(2) ßa( 2) /2
pF--pcF--p — F
dt dt 2 dt
A du(1) h da(1) 1 du(1)
pf~^+^Pf~---- PIv
dt
2
2c2
p!y
da
(1)
dt
■ + pF-
, du
(2)
dt
h 77
---2 pF -
2
da
(2)
dt
+
^v
du
(2)
dt 2c2
dt c-
da(2) pIy a
y dt
Su(2)
= 0
da(1)
„ du(1) h r du(1) , da(1)
pcF--+ p — F--+ pi--+ pc F
dt 2 dt y dt dt
h r da(1) h ^ da(1) + pc — F-+ p — cF-+
2 dt 2 dt
h' da(1) +p — F-+
4 dt 4
1 (h
r du(1) h ^ da(1)
pF-+ — pF-+ ,
2 dt 4 dt 4c2
h2 T da(1) h . du(1) ph-;- + ph — +
h _ du(2) hh Z7da(2) h du(2) hh
+ — pF 2 dt
,(2)
4
pF
dt
„ duh 7—' du(2) T da12) 2 „ dal/J h t-
-pcF--p — F--hp/--hpc F--hpc — F
dt 2 dt y dt dt 2 dt
2c2
,(2)
pK
y dt 2c
da(2) 1
dt
dt
4c'
-pK
,(2)
dt da
J.
(2)
Sa(1)
= 0
h da
(2)
+ pc—F 2
dt
- +
h2 da
+p—F-
4 dt
(2)
1
+ — 4
f
h
du(1) hh
^^^ pF 2 dt
4
pF
da
(1)
dt
+
h du(1) hh
2? p v ^T + 4?"
pi,
da
(1)
dt
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v
c
t
v
2
c
t
MEXANIKA
— ^du(2) — ^da(2)
—2 pF--I-—2— pF- pi , pi
2 dt 4 dt 2c2 y dt 4c2 2 dt
— , du(2) da(2) 1
2 p^^+'tTP1
J
dw1 1 pF—1 + — dt 4
, dw.
1 5w,
dw
pF ^^ + p— + pF-p—i
1 Sw,,
dt
dt
^ dw 1 f ^dw-pF—2 + -dt 4
1 dw.
dt
dw
c2 * dt 1
Sa
Sw,
(2)
= 0
= 0
dt c
Tabiiy umumlashgan chegaraviy shartlar:
2
Sw~
'{W + 2 MS) + i1cM!f) J+V 7(1)+2 71"+iM3' 1
_VN(1!)+1 MS' - j-Mi? 1+f7(22)+27(3) - Im'3 J
- (m + aNt! + N™ + — N(3 + — M,(,3) 1 +
(1)
Su
Su(2)
= 0
= 0
= 0
(13)
(14)
'U 2 4 11 4c 11
i , ,,(1) "TOO — -GO — -(3) — (3)
+|-Mf() + c 71 71 +f 71 + ^ 7
Sa
(1)
= 0
-^-M1(12) -cM1(12) -1M(12) -Mi3) + ^M1(13) j +
f »^(2) 2?(2) K -(2) ^ -(3) -..(3)'
+I^M71(2) - ( -y 7( 7( 7
-11 Ö1(33)+1 M1(33) J+i731)+1733)+^ M73(3)
-12 ö13) - J+f 7 32)+27 33) - ¿M
(3)
Sa
Swj
Sw,
(2)
= 0
= 0
= 0
Hisoblash algoritmi: Fazoviy statik va dinamik yuklangan sterjen tenglamalari (12), (13) va (14) hisoblash algoritmini ishlab chiqishda ularning vektor ko'rinishda quyidagicha keltirish mumkin.
Sterjennnig tebranish tenglamasi:
dt2 dx2 Tabiiy boshlang'ich va chegaraviy shartlar:
~dUk
[M ]
d t
dx
= 0,
m^]ük+[D]FclK=0,
(15)
(16)
(17)
Tenglamaning vektorli ko'rinishi (15), (16), (17) formulalarda M,A,B,C,D,M,B matrisasalardir [11-12].
Hisoblash algoritmini ishlab chiqishda chekli ayirmalar usulining oshkormas sxemasidan foydalanib tenglamani quyidagi ko'rinishga keltiramiz [13]:
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U/-XAX/K.!
Uch diagonalli tenglamalar tizimi (18) ifodani matrisali haydash usuli orqali quyidagicha
yozib olamiz:
UiJ+1=äMUMJ+1+ßM,
Bu yerda CCi+l va Д+1 noma'lum koyeffisiyentlar quyidagi ko'rinishdagi
В
"i+l =
; Д+1 = Aßl~ ; (i = i,2,...,/i-i).
Д • а, - С,
Л-а: -() . . .
formulalar yordamida topiladi.
Ikki tomoni qattiq mahkamlangan uch qatlamli sterjenga 50 kN kuch ta'sir qilgandagi deformatsiyalanganligini tekshiramiz. Uzunligi l=2000 mm, eni b=200 mm va balandliklari h1=5mm, h2=5mm, h3=190 mm o'lchamlarga teng.
1-rasm. X o'qi bo'yicha ko'ndalang 2-rasm. Z o'qi bo'yicha ko'ndalang kesim
kesimning maraziy o'qiga nisbatan umumiy makazuy o'qiga nisbatan umumiy ko'chishi ko'chishi
Xulosa. Fazoviy yuklanishlardagi uch qatlamli sterjen masalalarini Ostrogradskiy-Gamilton tamoyili asosida tuzulgan matematik modelning sonli natijalar olishda matrisali haydash usuli boshqa usullardan bir muncha aniqligi hamda qulayligini ko'rish mumkin.
ADABIYOTLAR
1. Кабулов В.К. Алгоритмизация в теории упругости и деформационной теории пластичности. - Ташкент: Фан, 1966. - 391 с.
2. Плескачевский Ю. Деформирование металлополимерних систем / Минск:Бел. навука., 2004. 342 с.
3. Старовойтов Э. Трехслойние стержни в терморадиационних полях. / Минск.: Беларуская наука, 2017. 275 с.
4. Anarova Sh.A., Yuldashev T. Derivation of Differential Equations of Oscillation of Rods in Geometrically Nonlinear Statement, Problems of Computational and Applied Mathematics. Tashkent, (2018). № 2. - Pp. 72-105.
5. С.П.Тимошенко, Дж. Гудьер — "Теория упругости"Перевод с английского М.Н.Рейтмана. М:. — "Наука" 1979.
6. Anarova Sh.A., Nuraliyev F.M., Usmonov B.Sh., Chulliyev Sh.I. Numerical solution of the problem of spatially loaded rods in linear and geometrically nonlinear statements.
7. International Journal of Engineering & Technology, 7 (4) (2018) 4563-4569.
8. Sh Anarova and Sh Ismoilov. Mathematical simulation of stress-strain state of loaded rods with account of transverse bending. 2019 J. Phys.: Conf. Ser. 1260 102002.
Mexanika va Texnologiya ilmiy jurnali 5-jild, 1-son, 2024
MEXANIKA
9. T.R. Rashidov, T.Yuldashev, D.A. Bekmirzaev. Seismodynamics of underground pipelines with arbitrary direction of seismic loading // Soil Mechanics and Foundation Engineering. Vol. 55. New York. 2018. Pp. 243-248.
10. Анарова Ш.А., Шокиров Д.А., Исмоилов Ш.М. Современное состояние и постановка задачи исследования трёхслойных стержней // Проблемы вычислительной и прикладной математики. - 2022. - № 4(42). - С. 54-78.
11. Анарова Ш.А., Шокиров Д.А., Исмоилов Ш.М. Современное состояние и постановка задачи исследования трёхслойных стержней // Проблемы вычислительной и прикладной математики. - 2023. - № 5(52). - С. 56-82.
12. Т. Юлдашев, А.И. Исомиддинов. Алгоритмы решения системы дифференциальных уравнений второго порядка и сравнительный анализ результатов // Узб. журнал «Проблемы информатики и энергетики».- Ташкент, 2011. - №2. - С. 29-34.
13. Ш.А. Анарова, А.И. Исомиддинов, Ш.М. Исмоилов. Численный анализ процессов колебаний пространственно-нагруженных стержней // Проблемы вычислительной и прикладной математики. - 2019. - № 5(23). - С. 29-44.
14. S. Anarova, S. Ismoilov and D. Shokirov, "Nonlinear Mathematical Model of Oscillation Processes of Spatially Loaded Rods with Account for Temperature," 2021 International Conference on Information Science and Communications Technologies (ICISCT), Tashkent, Uzbekistan, 2021, pp. 1-5, doi: 10.1109/ICISCT52966.2021.9670072.
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