Kwa kugwedezeka chifukwa cha kupinda, ndodoyo pa kusintha kwa kufanana imapinda mu ndege imodzi yayikulu.
Koma pali nawonso milandu ya kukhazikika imene ndodo, kapena mtanda, pakusinthika kwa kusamala zimakhala osati kungopindika kokha komanso kupindika mokhota (chith. 60a). Kwa ma support chochitika ichi chimatchedwa kugwedezeka kwa mbali, kwa ndodo zopanikizidwa zomwe pakugwedezeka zimakhudzidwanso ndi torsion, amati kugwedezeka ndi torsion.
Pa zochitika zonse ziwirizi, kukula kwa kulimba polimbana ndi torsion C, kapena kukana kupindika C*, n’kofunika kwambiri. Choncho, kusintha kwa kufanana komwe kumagwirizana ndi torsion kungakhudzidwe koyamba ndi njira zokulitsa kulimba polimbana ndi kupindika (maonekedwe oyenera a mtanda, mafelemu a m’mbali, kuthandiza kuchokera m’mbali ndi zina zotero). Kumbali ina, ndodo zooneka ngati tepi zokhala ndi mtanda wotseguka zimakhudzidwa kwambiri ndi kupindika pa torsion.
Msonkhano wa kusiyana kwa ma equation a kupindika m’mbali
Chith. 1a
Tiyeni mtengo wolunjika ukhale ndi ma moment a inertia okhazikika a kukula kosafanana kwambiri Ix >> Iy. Katunduyo uli mu ndege ya y-z-motero mtengowo udzapindika mu ndege yayikuluyi (u=ϑ=0). Zochitika komanso chiphunzitso zimasonyeza kuti malo amenewa a kulemera poyamba amakhala okhazikika, koma kwa katundu wokulirapo amakhala osakhazikika. Malo amenewa achiwiri a kulemera amalumikizana ndi kupindika kwa m’mbali mu u ndi ndi torsion ϑ.
Pansipa adzafotokozedwa chiyanjano cha kusiyanasiyana cha kupindika m’mbali. Ma equation a u ndi ϑ ndi a homogeneous. Ngati ma condition a m’mphepete alinso a homogeneous ndiye kupindika m’mbali kwa mtengo kungatheke kokha pa ma eigenvalue a vutolo, ndiye kuti u ndi ϑ zimatsalira koyamba pa ziro pa mtengo wofotokozedwayo wopanda kusokonezedwa. Chikhalidwe cha kupindika m’mbali chimapezeka – mofanana ndi kupindika kwa ndodo – ngati determinant ya ma coefficient a ma equation a homogeneous a ma condition a constants a kuphatikiza ikayikidwa kukhala zero. Tiona kuti ngakhale pa kupindika m’mbali pali mfundo yeniyeni ya nthambi ya kufanana kwa elastic.
Chith. 1b
Pa chith. 1b yasonyezedwa chigawo cha mtengo chotalika dz. Zotsatira za mikangano N1, Q1, M1, komanso kupindika κ__1 zimadalira katundu ndipo ndi zinthu zomaliza. Q, M, MD ndi ziro pa malo oyambirira a kufanana, ndipo zidzakhala zosiyana ndi ziro kokha pamene kupinda kopingasa kuyamba. Miyezo yomaliza, yomwe ikukhudzana ndi kuyamba kwa kupinda kopingasa, ndi yaying’ono kwambiri, monganso kusintha koyenera κ ndi d__ϑ/dz.
Magawo onse awiri omaliza a element dz ali mu ndege yowongoka pa axis yopindika kawiri ya ndodo ndipo adzapotoloka mu ndege izi zoyongoka.
Popeza ife – monga mmene zilili pa kuvutika kwa ndodo – chofunika kwambiri ndi kukula kwa katundu wovuta, ndiko kuti kuyamba kwa kupindika kwa mbali, osati kwambiri kukula kwa ma deformation, titha kunyalanyaza zinthuzo za kukula Q, M, MD ndi κ kapena ϑ’ monga kukula kachiwiri kakang’ono. Choncho, kuti tifotokoze bwino kwambiri, tidzaganiziranso kuti kupindika kwakukulu κ1 ndi kochepa kwambiri kwambiri ndipo kudzachitidwa ngati κ ndi ϑ’.
Chidziwitso: Chikoka cha κ__1 chiyeneranso kuwunikidwa mwaukadaulo ngakhale pa nkhani yapadera yosavuta (m’pamene M yokhala ndi mphindi yokhazikika ya kupinda ili ngati katundu). Pa zofunikira zambiri za luso la zomangamanga, κ__1 inganyalanyazidwe. Tingaganizire, mwachitsanzo, ngati kuti mtengowo pa nkhani ya katunduyo unapatsidwa kukweza kotere kuti pa katundu wovuta ndendende κ__1=0.
Kuchokera ku kulinganizika kwa zigawo za mphamvu m’njira za ξ, η, ζ ndi ma condition a ma moment pa ma axes ameneŵo, pambuyo pa zonyalanyazazi zomwe tazitchula, timapeza
Q’ + N1u’’ – Q1__ϑ’ + pϑ = 0_,_ (1a)
Q1’ + p = 0_,_ (1b)
N1’ = 0_,_ (1c)
M1’ – Q1 = 0_,_ (1d)
MD’ + Q + M1__ϑ’ = 0_,_ (1e)
MD’ – M1u’’ + pe ϑ = 0_._ (1f)
Kuchokera pa ma equation (1c, d ndi b) kumapezeka
N1 = const.,
Q1 = M1’, (2)
p = - Q1’ = - M1’’.
Pothetsa Q1 kuchokera mu eq. (1a) timapeza
Q’ = (M1’ ϑ_)’ – N1u’’_. (3a)
Kuchokera ku ndi. (1e) Q idzachotsedwa motero, kenako u ’’ kudzera mu u’’ = (MD’ + pe ϑ_)/M1_
(M+M1 ϑ_)’’ – N1/M1_ ∙ (MD’ + pe ϑ_) =_ 0_._ (3b)
Pofuna kupeza equation ya differential ya ϑ zidzayambitsidwa maubwenzi otsatirawa pakati pa ma moment ndi kusokonekera
M1 = -B1_∙v’’,_ (4a)
M = +B_∙_u’’, (4b)
MD = +C_∙ϑ’ – EC*∙ ϑ’’’,_ (4c)
kumene B1=EIx ndi B=EIy zimasonyeza kuuma pa kupindika kwa chonyamulira, C kuuma pa kupindika kwauzitsi, ndipo C* kukana kupindika. M’mawerengedwe athu otsatira tidzasiya mawu okhala ndi C* kuti zinthu zikhale zosavuta. Ngati ma equation akuyenera kugwiritsidwa ntchito mwachitsanzo pa ma I-profile, ndiye kuti ayenera kuwonjezeredwa moyenerera. Kunyalanyaza mawu v’’ kumatanthauza chimodzimodzi ndi kulingalira kuti B1=∞.
Ngati M ndi MD zitathetsedwa pogwiritsa ntchito nj. 4 timapeza equation yosiyanitsa ya kupindika kwa mbali kwa mtengo wa gawo lokhazikika
Chiganizo chosiyanitsa (5) ndi cha homogeneous ndipo ndi cha dongosolo lachinayi malinga ndi ϑ. M1 ndi ntchito yodziwika ya z malinga ndi katundu.
Gawo lokhala ndi momenti yosasintha ya kupinda
Tiyeni p=N1=0, kuphatikiza apo Mx=ɱ=const. Gawo lilibe flange (lophatikizika) ndipo giridi ndi lalitali poyerekezera ndi miyeso ya chigawo chopingasa, kotero tinganene kuti C*=0. Kenako ndi. (5) limakhala losavuta, chifukwa cha M1=ɱ, choncho:
ϑ’’’’ + ɱ2/BC∙ ϑ’’ = 0, (6)
kapena ndi chidule λ2_=ɱ__2/BC_
ϑ’’’’ + λ2 ϑ = 0. (6a)
a) Mtsenga wokhala ndi chithandizo cha mphanda pa malekezero onse
Chith. 2
Kuthandizira pa z=0 ndi pa z=l kuli ngati maforoko (chith. 2), amene amalola kupindika kwa u ndi v, koma amaletsa kutembenuka kwa magawo omaliza; ϑ=0. Popeza m’njira ya x palibe kukakamizidwa (i.e. M=0, kapena malinga ndi eq. (4) komanso u’’=0), zimatsatira kuchokera ku eq. ya u’’: MD’=0 ndipo potero kuchokera ku eq. (4c) ngati chikhalidwe chachiwiri cha malire ϑ’’=0.
Yankho lonse la equation (6a) ndi
ϑ(z) = A1∙sinλz + A2∙cosλz + A3∙λz + A4. (7)
Kuchokera ku zinthu za malire kumapezedwa
Dongosolo limeneli la ma equation ofanana lili ndi:
- Yankho losavuta A1 = A2 = ∙∙∙ = 0, limene likugwirizana ndi ϑ=0, u’’=0 kapena u=0, pomwe mtengo sukupindika pambali.
- Mayankho aakha malinga ndi Δ=0 kapena sin λ__l=0. Mizu ya chikhalidwe ichi cha kupindika pambali ndi λ__l=n__π (ndi n=1,2,3,∙∙∙) ndipo katundu wofanana woyambitsa mphindi
ɱ__K = _√_BC ∙ _n__π/_l. (8b)
Nthawi yocheperako ya kupindika m’mbali
min__ɱ__K = _√_BC ∙ π/l (8c)
zimachitika pa n=1. Yankho loyenera limapezeka pogwiritsa ntchito eq. (1f):
ϑ(z) = A1 ∙ sin πz/l (8d)
u(z) = C/ɱK ∙ ϑ(z).
b) Nkhani zina za kupindika kwa mbali
c) Kupindika pambali chifukwa cha momenti ya kupinda ndi mphamvu yopanikiza
Pa chitsanzo chowonetsedwa pa chith. 3 pali N1=-D (kukanikiza), p=0, ndiye C*=0_ yayikidwa. Pamenepo chilinganizo chosiyanitsa (5) chimakhala chosavuta ndipo chimati
ϑ’’’’ + λ2 ϑ’’ = 0, (9)
kumene λ__2 = ɱ__2_/BC + D/B_. (9a)
Yankho la differential equation (9) laperekedwanso ndi eq. (8), koma pano λ ikuyimira mtengo wina.
Chith. 3
Kwa chitsanzo cha chith. 3 zikugwiranso ntchito monga pa a) mikhalidwe ya contour ϑ=0 ndi ϑ’’=0 pa z=0 ndi z=l. Choncho ma equation a mikhalidwe pa A1 ali mofanana komanso mkhalidwe wa kupindika kumbali Δ=sin λl=0. Katundu wofunika kwambiri umakwaniritsa ubale λl=nπ (n=1,2,3) ndipo mtengo wochepa ndi
ɱK2/BC + DK/B = (π/l)2. (10)
Kwa D=0 kuchokera pamenepo pamapezeka momenti ya kupinda kwa mbali malinga ndi kuli. (8c) ndipo pa katundu wokakamiza wokha (ɱ=0) mphamvu ya Euler ya kupinda _DK=B(π/l)_2.
Pa chith. 3 nkhani yowonetsedwa iyenera kutchedwa kupindika kwapakati pa ndege y-z. Nkhani ya kukhazikika yomwe takambiranayi kale ndi kusokonekera kwa kunja kwa ndege ya kupindika komwe kumalumikizana ndi kupindika kwam’mbali u ndi torsion ϑ.
Ngati mbali zonse ziwiri zili zokhomeredwa (ϑ=0 ndi ϑ’=0) ndiye kuchokera ku mikhalidwe ya malire, pogwiritsa ntchito Δ=0, timapeza chikhalidwe cha kupindika kwa mbali
λ__l ∙ sin λ__l = 2(1-cos λl), (11)
ndi mizu λ__l=m__π (m=2,4,∙∙∙) ndi mphamvu za kupindika kwa mbali
ɱ__K__2/BC + DK/B = (2_π_/l)2.