Ga ƙarkatarwa yayin lankwasawa sandar tana lankwashewa a cikin wata babbar fanni ɗaya a lokacin canjin daidaito.
Amma akwai kuma irin waɗannan lokuta na kwanciyar hankali inda sanda, ko kuma ganda, a lokacin sauyin daidaito ba kawai suna lanƙwasawa ba har ma suna karkacewa (duba s. 60a). A cikin ɗaukar nauyi, ana kiran wannan al’amari lanƙwasawar gefe, kuma a cikin sandunan da aka matsa, waɗanda a lokacin lanƙwasawa kuma ake fallasa su ga karkacewa, ana magana da lanƙwasawa tare da karkacewa.
Ga duka abubuwan biyu, girman taurin da ke hana juyi C, wato juriya ga lankwasawa C*, yana da muhimmiyar rawa. Saboda haka, ana iya rinjayar canjin daidaito da ke tattare da juyi da farko ta matakan da ke ƙara taurin da ke hana karkatarwa (siffar ƙarshen giciye da ta dace, firam ɗin gefe-ƙetare, goyon bayan gefe da sauransu). A gefe guda, sanduna masu siffar rubutu tare da buɗaɗɗen ƙarshen giciye suna da matuƙar saurin lankwasawa lokacin juyi.
Daidaitattun lissafi na karkacewa ta gefe
Hoto 1a
Bari doguwar sanda madaidaiciya ta kasance da manyan moments na inertia masu tsananin banbanci Ix >> Iy. Lodi yana a cikin saman y-z-, don haka sandar za ta lanƙwasu a wannan babban saman (u=ϑ=0). Kwarewa da kuma ka’ida suna nuna cewa wannan matsayi na daidaito da farko yana da kwanciyar hankali, amma ga manyan lododi yana zama mara kwanciyar hankali. Wannan matsayi na biyu na daidaito yana da alaƙa da lanƙwasawa ta gefe a u da kuma torzija ϑ.
A ƙasa za a fitar da daidaitacciyar ƙa’idar bambanci don lankwasawar gefe. Ƙa’idodin u da ϑ na homogene ne. Idan kuma yanayin iyaka na homogene ne, to lankwasawar gefen katako zai yiwu ne kawai ga ƙimomin kai-tsaye na matsalar, wato u da ϑ sukan fara zama sifili ga katakon da aka bayyana a sama ba tare da wata hargitsi ba. Ana samun yanayin lankwasawar gefe - kamar yadda ake yi wajen lankwasawar sanda - idan aka ɗauka cewa determinant ɗin ƙa’idodin sharadi na homogene don lambobin haɗawa daidai yake da sifili. Za mu ga cewa ko a lankwasawar gefe akwai tabbatacciyar mahadar rabuwar daidaiton elastik.
Siffa 1b
A hoto. 1b an nuna ɓangaren katako mai tsawon dz. Sakamakon damuwa N1, Q1, M1, da kuma lankwasa κ__1 suna dogara da nauyi kuma suna da iyakantacciyar girma. Q, M, MD sifili ne a matsayen daidaito na farko, kuma za su bambanta da sifili ne kawai idan lankwasa ta gefe ta fara bayyana. Iyakantattun girma, waɗanda suka shafi farkon lankwasawar gefe, ƙanana ne matuƙa, haka nan kuma canje-canjen da suka dace κ da d__ϑ/dz.
Duka ɓangarorin ƙarshe na ɓangaren dz suna kwance a cikin yanayin al’ada zuwa ga axis mai lanƙwashewa biyu na sandar, kuma za su kasance a lanƙwashe a cikin waɗannan wuraren al’ada.
Tun da mu – kamar yadda yake a wajen lanƙwasawar sanda – a matakin farko abin da ya fi ba mu sha’awa shi ne girman nauyin kritikal, wato farawar lanƙwasawar gefe, ba girman nakasu ba, za mu iya yin watsi da ninkewar ƙimomin Q, M, MD da κ ko ϑ’ a matsayin ƙananan ƙimomi na mataki na biyu. Saboda haka, domin ƙarin sauƙi, za mu kuma ɗauki babban lanƙwasa κ1 a matsayin ƙarami ƙwarai kuma za a mu’amala da shi kamar κ da ϑ’.
Lura: Za a iya kimanta tasirin κ__1 a ka’idance ma a cikin sauƙaƙen yanayi na musamman (moment na lankwasawa M a matsayin nauyi na dindindin). A mafi yawan bukatun fasahar gini ana iya yin watsi da κ__1. Misali, za mu iya ɗauka kamar gindin ya kasance an ɗaga shi sosai ga wannan yanayin na nauyi ta yadda ga nauyin mai mahimmanci daidai κ__1=0.
Daga daidaiton abubuwan ƙarfi a kwatance ξ, η, ζ da kuma sharuɗɗan moment dangane da waɗannan axises ɗin, bayan abin da aka ambata na ragewa, ana samu
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)
Daga eq. (1c, d da b) ana samu
N1 = const.,
Q1 = M1’, (2)
p = - Q1’ = - M1’’.
Ta kawar da Q1 daga ƙa’ida (1a) ana samun
Q’ = (M1’ ϑ_)’ – N1u’’_. (3a)
Daga jedn. (1e) za a kawar da Q ta wannan hanya, sannan u ’’ ta hanyar u’’ = (MD’ + pe ϑ_)/M1_
(M+M1 ϑ_)’’ – N1/M1_ ∙ (MD’ + pe ϑ_) =_ 0_._ (3b)
Domin a samu ƙa’idar bambanci ga ϑ za a gabatar da waɗannan dangantaka tsakanin lokutan juyi da nakasu
M1 = -B1_∙v’’,_ (4a)
M = +B_∙_u’’, (4b)
MD = +C_∙ϑ’ – EC*∙ ϑ’’’,_ (4c)
inda ke nufin B1=EIx da B=EIy tsaurin lanƙwasawar katako, C tsaurin karkatarwa, sannan C* juriyar murɗawa. A cikin ƙarin lissafinmu za mu bar kalmomin da ke da C* don sauƙi. Idan za a yi amfani da daidaitattun misalan ga I-profila, to wajibi ne a ƙara su yadda ya dace. Yin watsi da kalmar v’’ yana nufin daidai da ɗaukar B1=∞.
Idan aka kawar da M da MD ta hanyar amfani da jedn. 4 ana samun lissafin bambanci na lankwasawar gefe na katako mai sashen giciye iri ɗaya
Ƙa’idar bambanci (5) tana daidaitacciya ce kuma ta mataki na huɗu dangane da ϑ. M1 sanannen aiki ne na z gwargwadon ɗaukar nauyi.
Katako mai ƙarfin lankwasawa na dindindin
Bari p=N1=0, ban da haka Mx=ɱ=const. Sashen ba shi da flange (mai matsewa) kuma katako dogaye ne idan aka kwatanta da girman sashen giciye, don haka za mu iya ɗauka cewa C*=0. Sai a sauƙaƙa jedn. (5), saboda M1=ɱ, don haka:
ϑ’’’’ + ɱ2/BC∙ ϑ’’ = 0, (6)
ko da ta ƙirƙirarwa λ2_=ɱ__2/BC_
ϑ’’’’ + λ2 ϑ = 0. (6a)
a) Ganga mai goyon bayan cokali a duka ƙarshensa
Fig. 2
Dogaro a wajen z=0 da z=l yana cikin siffar cokula (sl. 2), waɗanda ke ba da damar lanƙwashewa u da v, amma suna hana juyawar sassan ƙarshen; ϑ=0. Tunda a cikin hanyar x babu kullewa (wato M=0, ko kuma bisa ga ma’a. (4) haka ma u’’=0), daga ma’a. na u’’: MD’_=0 ake samu, don haka daga ma’a. (4c) a matsayin sharadin iyaka na biyu ϑ’’=0.
Mafita ta gaba ɗaya ta daidaituwar lamba. (6a) ita ce
ϑ(z) = A1∙sinλz + A2∙cosλz + A3∙λz + A4. (7)
Daga sharuddan kan iyaka ana samu
Wannan tsarin daidaitattun ƙa’idodi yana da:
- Maganin sauƙaƙe A1 = A2 = ∙∙∙ = 0, wanda ya dace da ϑ=0, u’’=0 ko u=0, inda katakon ba ya karkata ta gefe.
- Maganganun kai tsaye bisa Δ=0 ko sin λ__l=0. Tushen wannan sharadin lankwasawar gefe su ne λ__l=n__π (da n=1,2,3,∙∙∙) da kuma madaidaitan lodin momenti na ƙarshe
ɱ__K = _√_BC ∙ _n__π/_l. (8b)
Mafi ƙarancin momentin karkacewar gefe
min__ɱ__K = _√_BC ∙ π/l (8c)
yana samuwa a n=1. Maganin da ya dace ana samu ta amfani da eq. (1f):
ϑ(z) = A1 ∙ sin πz/l (8d)
u(z) = C/ɱK ∙ ϑ(z).
b) Sauran shari’o’in karkacewar gefe
c) Lanƙwasawar gefe sakamakon lokacin lankwasawa da ƙarfin matsawa
Ga misalin da aka nuna a sl. 3 an N1=-D (matsi), p=0, sannan aka sa C*=0. Sa’an nan daidaiton bambanci (5) ya sauƙaƙa kuma ya zama
ϑ’’’’ + λ2 ϑ’’ = 0, (9)
inda λ__2 = ɱ__2_/BC + D/B_. (9a)
Maganin ƙa’idar bambancin (9) an sake bayar da shi a cikin eku. (8), sai dai yanzu λ yana nufin wata daraja ta daban.
Hoto. 3
Ga misalin sl. 3 suna aiki kamar na a) ka’idojin iyaka ϑ=0 da ϑ’’=0 ga z=0 da z=l. Saboda haka ƙa’idodin lissafin sharaɗi na A1 ma suna daidai da formal kuma haka ma sharaɗin lanƙwasawar gefe Δ=sin λl=0. Nauyin da ke da muhimmanci yana cika dangantakar λl=nπ (n=1,2,3) kuma mafi ƙanƙantar ƙima ita ce
ɱK2/BC + DK/B = (π/l)2. (10)
Ga D=0 daga nan ana samun moment ɗin lankwasawar gefe bisa ga ma’auni. (8c) kuma ga nauyin matsawa tsantsa (ɱ=0) ƙarfin lankwasawar Euler- na _DK=B(π/l)_2.
A sl. 3 yanayin da aka nuna ya kamata a kira shi lankwasawa mai karkata a cikin fagen y-z. Hali na kwanciyar da aka yi nazari a baya, saboda haka, karkacewa daga fagen lankwasawa ce da ta haɗu da lankwasawa ta gefe u da jujjuyawa ϑ.
Idan duka ƙarshen sun kasance an kulle (ϑ=0 da ϑ’=0) to daga yanayin iyaka, ta hanyar Δ=0, ana samun sharadin lanƙwasawar gefe
λ__l ∙ sin λ__l = 2(1-cos λl), (11)
da tushen λ__l=m__π (m=2,4,∙∙∙) da kuma ƙarfin lanƙwasawar gefe
ɱ__K__2/BC + DK/B = (2_π_/l)2.