** Archive nyanzvi yemukati: ** zvinyorwa uye mafomula zvinoburitsa nhoroondo yetioretical mharidzo uye haisi purojekiti, static calculation, inotakura-inotakura kugona kutarisa, kuongorora kwechinhu chiripo, kana mirairo yekuvaka kana kugadzirisa. Dhizaini, fungidziro uye fungidziro mapatani anofanirwa kutariswa zvichienderana neiyo chaiyo sisitimu, zvinhu, geometry, mamiriro emuganho uye anoshanda zviyero. Kuverengera torsion, kugadzikana, kuneta, majoini uye kushushikana nekuda kwekudzivirirwa kukotama kunofanirwa kuitwa neinjiniya ane rezinesi, ine yakazvimirira kutonga kana ichidikanwa.

Nhasi, Savo Kusić inotarisa pa mahwindo emapuranga, wood-aluminium Mahwindo, mahwindo etsika, gonhi uye zvikumbiro zvekotesheni Ichi chinyorwa chinoramba chiri chenhoroondo nyanzvi yekuchengetera zvinyorwa uye haimiriri basa rekugadzira dhizaini.

Saint-Venant dzidziso ye torsion (yemahara torsion)

Differential equation

Cantilever tsvimbo yakatwasuka padivi pe x-axis yakatakurwa nekanguva kekurovera M pamagumo emahara

Sl.1

Ita kuti tsvimbo yakatwasuka igadzirwe kune imwe yemagumo ayo x=0uye ita kuti itakurwe kune imwe mugumo nechinguva chekushungurudza M (fig.1) Zvadaro chimwe nechimwe chezvikamu zvaro chinotenderera neimwe kona α(x) ichitenderedza imwe mutsara unofambirana neakisi yetsvimbo, yatinozotora se x-axis. Kana zvikamu zvese zvakaenzana kune mumwe nemumwe, chero mutsara wakatwasuka wakafanana ne-x-axis uchapfuura mucoil, uye kona α ichave yakaenzana nechekure kubva kumagumo akatarwa: α=ϑx, kubva painosimuka chikamu mundege yechikamu (Fig.2):

v = - r chivi__ψ = - ϑ x z,

w = r a cos__ψ = ϑ x y.  (1a)

Huwandu ϑ hunodaidzwa kuti torsion angle (dimension: kureba-1, kutenderera pakureba kwechikamu). Iyo torsion angle inoenzanirana nenguva yekushungurudza M:

ϑ = M/GJT.   (2)

Muchinjikwa chikamu chetsvimbo ine y uye z coordinates, radius uye shear stress components

Fig. 2

The proportionality factor GJT, torsional stiffness, yakaenzana nechigadzirwa cheshear modulus G uye factor JT, inomiririra kushanda kwechimiro nekukura kwechikamu.

Uku kutenderera kwechikamu hakusi iko kwega deformation yetsvimbo. Mukuwedzera, kunewo kupeta kwechikamu-chepakati, kureva kusimuka u munzira ye x-axis, iyo ine hukuru hwakafanana hwezvikamu zvinopindirana zvezvikamu zvose zvepamuchinjikwa uye naizvozvo inongova basa rezvirongwa mundege yechikamu, uye neimwe nzira inoenderana nemutoro Mion

u = u(y,z) = ϑ ∙ φ(y,z), dφ/dx=0.  (1b)

Kana mataurirwo ari pamusoro akaunzwa mune anofambirana equation equation, zvinoonekwa kuti tsungiriro huru dzakaenzana dzakaenzana ne zero, zvichisiya zvisungo zviviri chete txy ne τxz:

Equations of shear stress components tau xy and tau xz inotaridzwa maererano nekutamisa uye warping function

Poshi.3

Iyo yekusiyanisa equation yedambudziko re torsion inowanikwa zvichienderana neiyo equations (3) kuisa:

Two Differential Balance and Compatibility Equations for the Torsion Problem

Poshi.4uye i4b

nekugadzirisa misiyano iyi yezvisingazivikanwe τxy uye τxz nekusuma basa rekusimbisa F(y,z) iro:

τxy = - dF/dz,   τxz = + dF/dy.  (5)

Neizvi, equation (4a) zvakafanana kugutsikana, uye kubva kune equation (4b) iyo inodiwa yekusiyanisa equation inowanikwa

d2F/dy2+ d2F/dz2=2G__ϑ (6)

Mamiriro emuganho anodiwa kune yakasarudzika mhinduro yeiyi equation anowanikwa kubva kune chinodiwa chekuti hapana kudzvanya kweshear pahamvuropu yetsvimbo, kubva iyo, nekuda kweiyo conjugate hunhu hwekuveura kushushikana, zvinotevera kuti kushushikana kuri muchikamu yendege hakufanirwe kuve nechikamu cheperpendicular kwairi pamwe neiyo contour. Kubva pano, fig.2:

dy/dz = _τxy/_τxz = - (dF/dz) / (dF/dy)

naizvozvo

dF/dy dy + dF/dz dz =0,

i.e. mutsauko wese webasa F unofanira kuva zero, kureva kuti, inofanira kuva F=const. pamwe ne contour.

Kana chikamu chiine contour imwe chete, chinogona kungoiswa F= pachiri0, nekuti nebasa remagetsi, iro isu tinenge tichingofarira chete zvakatorwa, iyo yekuwedzera yekuwedzera isingaite basa. Kune zvikamu zvine zvinopfuura contour imwe, nyaya iyi yakaoma.

Sezvinoonekwa kubva ku equation (5) mhedzisiro yeshear stress τxy ne τxz panzvimbo yemharadzano yakaenzana yakaenzana muhukuru ne gradient yestress function F, asi iri perpendicular kwairi, i.e. ine tangent direction kumutsara F=const.

Nekugadzirisa mutsauko equation (6)) magetsi τ semabasa e y, z uye ϑ anowanikwa. Basa rekuverenga nderokuti, kugadziriswa kwekona ye torsion ϑ uye kushushikana, kunyanya kunyanya kunetseka sekushanda kwechinguva chekuvhiringidza, saka, kugadziriswa kwekuoma kwechikamu JT kubva pakuenzanisa (2) uye torque yekupikisa WT = Mmaxτ. Kuti uwane izvi, zvinongodiwa kutaura M sebasa re F:

Integral expression of the torsional moment over stress components and stress function F

Zvese zviri zviviri zvirevo zveiyi yakakosha, iyo inofanirwa kutorwa pamusoro pechikamu chose, inogona kushandurwa nekubatanidzwa kwechikamu. Kana, semuenzaniso, yekutanga yadzo yabatanidza kutanga pamutsara z=const. kubva panzvimbo ye contour y=y1 kusvika padanho y=y2 (mu.3) uye tarisa kuti mune zvese izvi zvibodzwa F=0, inowanikwa

Kubatanidzwa muchidimbu kwezwi nerinobva pakuita kwekushushikana mupfungwa F pakati pemiganhu y1uye y2

Iko kukosha kwakafanana kunowanikwa kubva kune yechipiri yakakosha, kuitira kuti

Pattern of torsion moment M seyakapetwa kaviri pamusoro pebasa rekushushikana F

Poshi.7

Hameno-chikamu-chikamu chine mutsetse wakachinjika pakati pemapoinzi y1uye y2

Sl.3

Dzimwe nyaya dzezvikamu zvizere

The differential equation (6) ndiyo inhomogeneous equation yezvinobvira dzidziso. Nokudaro, nokuda kwekugadzirisa dambudziko rekutambudzika munyaya ye-arbitral cross-section, nzira dzezvinogona kuitika dzidziso dziripo, dzinoita kuti igare ichiwanikwa. Idzi nzira, zvisinei, kazhinji dzakanyanya kuoma kuti dzishandiswe kune yega nyaya dzinomuka mukuita. Nokudaro, tafura inotevera inoratidza migumisiro yekuverenga kwakadaro kwezvikamu zvakawanda.

Table yekuomarara zvinhu uye torsional kuramba nguva yakasimba uye isina hollow denderedzwa, elliptical uye rectangular zvikamu

Kune zvidimbu-zvidimbu zviri pedyo nedenderedzwa, JT=F inogona kuiswa4/40_Ip_, apo F inzvimbo uye Ip inguva yepolar yeinertia yechikamu.

** Cherekedza pane mafomula nematafura: ** matauriro uye kukosha mumifananidzo yakaongororwa zvinochengetwa senhoroondo sosi. OCR zvinyorwa, index mamakisi uye scan haina kuvimbika zvakakwana pakuverenga. Usati washandisa, imwe neimwe fomula, yuniti, coefficient uye nharaunda yechokwadi inofanirwa kuongororwa mune inotenderwa nyanzvi sosi uye inoshanda chiyero.

Kufananidza ne membrane

Rubatsiro rwakanaka rwekufungidzira torsional stress, uye panguva imwe chete nzira yekutsunga kwavo kwakaringana inopiwa nekuenzanisa pakati pebasa rekushushikana F yedambudziko rekurovera uye deflection transverse mutoro we membrane. Ngakuve neavhareji yekuvhura pamadziro akafuratira emudziyo anopindirana nechikamu chemuchinjiko wetsvimbo yakatakurwa mu torsion. Kana iyo membrane yakatambanudzwa pamusoro pekuvhura uku uye kudiki kudiki kudzvinyirira p kwakonzerwa mumudziyo, membrane inobuda kunze, zvekuti kutsauka kwayo ξ(y,z) kugutsa musiyano we equation.

Differential equation yekutsaudzira kwebrain-loaded membrane

apo N isimba repamusoro mu membrane rinokonzerwa ne capillary forces (dimensions: force per unit urefu). Equation iyi inopindirana neiyo equation (6), uye mamiriro emuganhu ξ=0yakafanana neiyo yemagetsi ekushanda. Nenzira iyi, iyo voltage function _F (y, z) _ inogona kuwanikwa uchishandisa iyi girazi ine membrane uye nekudaro kugadzirisa dambudziko retorsion. Kushushikana τ kunoenderana nemateru e membrane maererano ne y-z-ndege, saka nzvimbo yakakwirisa ndiyo iyo danda rakafumurwa kune torsion rine yakanyanya kushear stress.

Geometry yechikamu cheI yakakamurwa kuita rectangles nhatu dzine denderedzwa rakanyorwa pamubatanidzwa wembabvu netsoka

Sl.4

Dhiagiramu yealpha coefficient zvinoenderana nereshiyo yezviyero zvezvikamu zvechikamu cheI

Sl.5

Iyi membrane maitiro anonyanya kukodzera kuwana maitiro ekufungidzira ezvikamu zvitete-madziro. Apo, semuenzaniso, I-chikamu chinoparadzaniswa maererano nemufananidzo.4pamarectangles matatu uye kune imwe neimwe yadzo fananidzo ine membrane inoitwa zvakasiyana, huwandu hwehuwandu hwezvakabuda protrusions huchave hudiki zvishoma pane vhoriyamu inonyatsoenderana nebasa remagetsi. Saka kana makobvu ari _b1_uye b2 diki, zvinenge

JT =2JT1+ JT2,

kupi JT1 uye JT2 kuomarara zvinhu zvemunhu rectangles anogona kuverengerwa zvichibva patafura1. Nzira iri nani inowanikwa kana JT yaiswa mbabvu2 = a2b32/3, saka, kana mbabvu ichitorwa sechikamu cherectangle yakareba kwazvo. Hukoshi hwakatonyanya kunaka hunowanikwa zvinoenderana ne Trayer uye March\ kubva

JT =2JT1+1/3a2b32+2 α D4,

apo D ndiro dhayamita redenderedzwa guru rakanyorwa sezvinoratidzwa mufig.4, uye α inogona kutorwa kubva muonde.5. Kune chikamu cheT chinowanikwa zvakafanana

JT = JT1+JT2+ α D4,

kupi JT2 hafu yemari yakaverengwa maererano netafura1zve a=2_a2_, b=b2. Kune L-zvikamu (Fig.6) zvakafanana nazvo

JT = JT1+ JT2+ β D4

kupi JT1 nekuti ruoko rwakakora runotorwa zvakananga kubva patafura1, apo JT2 yegumbo rakatetepa rinoverengerwa sechikamu cheT, uye β inotorwa kubva muonde.7.

L-chikamu chejiometri ine makumbo mativi, ukobvu uye radius yekutenderera

Fig. 6

Dhigiramu yebeta coefficient zvichienderana nereshiyo yezviyero zvechikamu cheL

Sl.7

Paunenge uchiverenga kushushikana, iyo torsional nguva yakakamurwa kuita constituent rectangles: iyo chikamu chinowira paflanges M.1=M JT2/JT; iyo shear stress inozoverengerwa pakati perutivi rwakareba rwerectangle yega yega zvichibva patafura1. Iyo yakakura yezvidzvanyiriri izvi haifanire kuve iyo yakanyanya kushear stress inoonekwa muchikamu. Sezvineiwo, kana kona yechiitiko isina kutenderedzwa, τ=∞ inowanikwa panzvimbo iyoyo, kuitira kuti kune diki diki diki yekutenderedza, yakanyanya kuchekwa stress inogona kutarisirwa pano.

Engineering Cherechedzo: the mathematical singularity of the idealized acute angle harisi rezenisi rekufungidzira iwo chaiwo chaiwo neiyi fungidziro pateni chete. Iyo chaiyo inotenderedza geometry, kushivirira, plasticity, welds, kusara kunetseka, kugadzikana kwenzvimbo uye kuneta kunoda kwakakodzera modhi uye ongororo yehunyanzvi.

Kune kona yedivi rimwe uko ukobvu hwerutivi t zvinoenderana ne Trefftz\ iri pakutenderedzwa kwedhayamita r:

Pattern of the maximum shear stress in cornering section kutenderera

kupi ne τ0 yakatemerwa huwandu hwemagetsi akaverengerwa gumbo rega rega zvichienderana nemaitiro anotsanangurwa pano. Kune yakakura kutenderera radii, τ= inogona kugamuchirwa2__τ0.

Mapaipi akatetepa akakomberedzwa (Bradt Pateni)

Mune mapaipi akatetepa-akavezwa, iyo inoputika membrane ine chimiro chinoratidzwa mufig.8, iyo inosanganisira ndege iri pamusoro penzvimbo isina mukati mukati mepombi uye nzvimbo yakatsetseka inoipoteredza uye inorara pamusoro pechikamu chinoshanda chemadziro epombi; nzvimbo iyi yakarerekera inofanira kunge yakakwira kana madziro epombi ari matete panzvimbo yakakodzera. Muchiitiko ichi, shear stress inogoverwa zvakaenzana pamusoro pehupamhi hwemadziro uye inopindirana nehupamhi hwemadziro, i.e. shear force T = τt haishanduki pamwe chete nedenderedzwa rechikamu.

Enzaniso yeMembrane yechikamu chakavharwa chine madziro matete ane nzvimbo yakati sandara uye yakarerekera pamusoro pemadziro epombi

Sl. 8

Iyo yepamusoro element t_∙ds_ inogamuchira chikamu chenguva yekutambudza

dM = t_∙ds∙τ∙h,_

saka nguva yose iri

Torsion moment pateni yechikamu chitete-chakavharika chakavharwa pamusoro pesimba rekugera nenzvimbo yakavharirwa

Poshi.8

apo nzvimbo yeFR yakafukidzwa nemutsara wepakati wakatarwa nemitsara uye madotsi mumuonde.8. Inowanikwa kubva pano

Pateni yekunyanya kudzvanya kweshear muchikamu chakavharwa chine madziro matete

Poshi.9- Yekutanga Bredt\ pateni

Kuomarara kweTorsional kunowanikwa zvakanyanya nyore kana basa rekushandura richiratidzwa kune rumwe rutivi kuburikidza ne torsional nguva uye torsion angle uye kune rumwe rutivi kuburikidza neshear stress. Nenzira iyi, inowanikwa kune tsvimbo yehurefu l

Equation yedeformation yebasa retsvimbo rinoratidzwa nechinguvana, torsion angle uye shear stress

saka, kuunza kukosha kwe τ maererano ne equation (8),

The deformation work equation mushure mekuzivisa matauriro aBredt ekuti shear stress

ndiko kuti, nekuenzanisa ne equation (2):

Matauriro aBradt ekuomesa musoro kwechikamu chakavharika chine madziro matete-ekusiyana ukobvu

Poshi.10

Kana ukobvu hwemadziro hwakafanana kumativi ese, saka zviri nyore (2. Bredt’s pattern):

Matauriro aBradt ekuomarara kwechikamu chakavharwa cheukobvu hwemadziro akaenzana

Poshi.11

Zvakajairwa kushushikana mutorsion nekuda kwekukotama kwechikamu

Matemu ekutanga uye musiyano equation

Component displacements u, equation (1b), inosarudza kukombama kwezvikamu kubva mundege dzavo. Kana nguva ye torsion ichigara, uku kukotama kwakafanana muzvikamu zvese uye zvakajairwa zvinonetsa σx hazvioneke.

Muzviitiko zvakawanda, kupeteka kwechikamu-chikamu, icho chinofanira kusimuka pahwaro hwe_St. Dzidziso yaVenant haigoneki. Kana iyo torsional nguva ikachinja kamwe kamwe muchikamu chimwe chete, kukotama kwehukuru hwakasiyana kwaizoitika pamativi ese emuchinjiko-chikamu, uye kuenderera kweiyo deformation kwaizokanganiswa. Imwe mugumo wetsvimbo inogona kusungirirwa zvekuti kukotama kwechikamu kunobviswa zvachose panguva iyoyo. Mumamiriro ezvinhu akadaro, shanduko inofanira kuitwa _St. Venant’s theory. Supplementary stresses σx mugwara reaxis yetsvimbo uye supplementary shear force T (kuveura zvinosimbisa T/t) zvinoonekwa mutsvimbo, iyo iri yepamusoro-soro muchikamu icho kuenderera kwe deformation kwakatyoka uye iyo inoderera exponentially nechinhambwe kubva pachikamu ichi. Panyaya yematanda akasimba uye machubhu, dzidziso iyi yakaoma, uye kuwedzera kwemagetsi kunoderera nekukurumidza zvekuti kutsunga kwavo hakuna kukosha muzviitiko zvakawanda. Kune matete-walled akavhurika-profile tsvimbo, kururamisa uku kwakakosha zvikuru uye kunogona kuverengerwa zviri nyore.

Mutsetse wepakati wechikamu chakavhurika chakatetepa-chakakomberedzwa chine marongedzero, pakati pekutenderera uye chinhambwe chakatetepa

Sl.9

Sl.9inoratidza mutsara unosangana nechepakati pemadziro uye ndege yechikamu inopindirana (mutsara wepakati wechikamu). O ndiro pakati pekutenderedzwa kwezvikamu zviviri zvakatarisana. Kana chinhambwe chavo chiri dx, kona ye torsion iri ϑ dx, saka nzvimbo isina kufanira yepakati inosvika pakubviswa u ϑ dx. Kutamiswa uku kune chikamu rt ϑ dx munzira tangent kune chikamu, uye ndiyo inotsvedza yechinhu chepakati pemadziro (tichifunga kuti _u=_φϑ):

γ=__ϑ (rt + d__φ/ds).

Sezvo shear stress iri pakati pemadziro iri zero, inofanirwa kunge iri γ=0, saka ndosaka

Integral expression ye curvature function fi pamwe nepakati pechikamu chakavhurika

Iyi pateni ine matatu anopokana maparameter: kwakabva uko kureba kwearc s kunoverengerwa uye marongero epakati pehukama kutenderera O, iyo rt inotsamira. Nzvimbo yekutanga kubva iyo arc kureba s yakaverengerwa inogona kusarudzwa kuitira kuti iyo yepakati curvature kukosha iite zero:

Mamiriro ezvinhu ekuti kukosha kwechirevo che curvature basa pamusoro penzvimbo yemuchinjikwa yakaenzana ne zero

uye zvinorongeka zvepakati O zvinogona kutariswa nenzira yekuti nguva dzakakonzerwa nekukotama kwechikamu ive zero:

Mamiriro maviri akakosha enguva ye zero ye curvature function zvinoenderana nezvirongwa y na z

Zvinogona kuratidzwa kuti center of rotation saka yakatemwa inopindirana neshear centre iri pano yakatsanangurwa uye yakatsanangurwa neimwe nzira.

Kana kukotama kwechikamu chinotsanangurwa nemapatani aya kwakadzivirirwa zvachose kana kuti zvishoma, zvinowanzoitika σx munzira yeaxis yetsvimbo inofanira kuoneka. Kunyatsoongorora kunoratidza kuti kushushikana uku kunogona kutorwa kuti kuenzane ne curvature function φ: σx=k__φ, apo φ inotsamira chete pa s uye k chete pa x. Maererano nemutemo waHooke:

σx = E_∂u/∂x = Eφ dϑ/dx_, (12)

inofanira kuva k=E d__ϑ/dx. Zvakajairwa zvinonetsa σx zvinodaidzwa kuti chikamu chekupeta kushushikana. Tichifunga nezvemaequation ari pamusoro, ivo vanoumba equilibrium system yemauto muchikamu chimwe nechimwe, hurongwa hwemasimba nekuda kwekukotama kwechikamu.

Iyo torque yakazara ine zvikamu zviviri: chimwe chazvo M1=GJT__ϑ inomuka nekuda kwekunetseka kwekugera kwakapihwa na_St. Venant_'s theory, uye imwe M2 inoitika nekuda kwekukotama kwechikamu:

Musiyano equation yenguva yakazara yekutenderera neSaint-Venant uye chikamu chakakomberedzwa

Poshi.13

Sector moment of inertia sechinhu chakakosha cheskweya yebasa rekupeta nekukora kwemadziro

Sezvo nguva M yapihwa sebasa re x, chirevo ichi chinomiririra equation yekusiyanisa yekona yekudzosa ϑ. Kana iyi equation yagadziriswa, _M inogona kuverengerwa1_uye M2, uye nekudaro zvese zvinonetsa uye madhiri.

Mhinduro nemashandisirwo

Kana iyo torsion nguva M ichigara, iyo general mhinduro ye equation (13) ndizvo:

General mhinduro ye torsion angle differential equation nekubatanidza zvinoramba zvichiitika C1uye C2

Integration constants C1, C2 inofanira kutariswa kubva pamiganhu miviri.

a) Yakasungwa zvakasimba kupera x=0(muonde.1)

Kumagumo aya u≡0saka zvichibva pane equation (1b) ϑ=0. Kana tsvimbo yakarebesa (αl>>1), ipapo C1=0uye c2= -M/GJT. Munyaya yetsvimbo pfupi, kutaridzika kwekunetseka nekuda kwekukotama kwechikamu kunokonzerwa nemamiriro ekupedzisira kwetsvimbo. Kana kukotama kwechikamu kukasadzivirirwa panzvimbo iyoyo, pachave ne σx≡0uye dϑ/dx=0. Inowanikwa kubva pano

Matauriro ezvekubatanidza zvinoramba zviripo C1uye C2yetsvimbo ine muganhu mamiriro

b) Symmetrical torsion load maererano neonde. 10

Kuruboshwe hafu yetsvimbo, nguva yekurovesa muviri ndeye -M/2, uye kurudyi +M/2. Muchiitiko chekusununguka kusununguka, zvikamu kuruboshwe uye kurudyi zvepakati petsvimbo zvingave zvakamonyoroka zvakatarisana, saka kuenderera mberi kwekugadzirisa kwaisazochengetwa. Naizvozvo, panzvimbo x=0, nguva yese yekurovera inofanirwa kutamiswa kuburikidza nekunetsekana kunoitika nekuda kwekukotama kwechikamu, uye nekudaro muchikamu chino M1=0 uye saka ϑ=0. Panguva ino, hurongwa hwemasimba hunoita kuti huve nechokwadi chokuti chikamu chinoramba chakatwasuka, kunyange zvazvo zvikamu zvakarara kure nepakati pemudonzvo wetsvimbo. Kana, nekuwedzera, zvinotarisirwa kuti pamagumo etsvimbo zvinonetsa σx≡0, imwezve muganhu wemuganhu unowanikwa kune imwe neimwe hafu yetsvimbo, saka kune hafu yakarurama iyo

Torsion angle solution yehafu ye symmetrically loaded rod

Tsvimbo yakasungirirwa kumagumo ese uye yakaremerwa nenguva yekudzura pakati

Fig. 10

c) External moment M pane imwe nzvimbo isingaite (fig. 11)

Pakadai, mhinduro ϑ yezvikamu zviviri zvetsvimbo inofanira kunyorwa zvakasiyana, saka zvikamu zvina zvinoramba zviripo C1, C2, C3, C4 zvinoonekwa. Pamusoro pezvo, nguva dzekutambudzika Ma, Mb hadzizivikanwe. Manhanhatu equation aripo: kuenderera kwechimiro che ϑ (inoratidzwa na u) uye ye dϑ/dx (inoratidzwa ne σx) panzvimbo yekuremerwa chiito, chimiro chimwe pamagumo etsvimbo (e.g. rigid pinching ϑ=0 kukotama, kusavhiringidza dϑ/dx=0), nguva yakaenzana mamiriro -Ma+Mb=M uye pakupedzisira mamiriro ekuti zvikamu zvekupedzisira hazvitendeuki kune mumwe nemumwe: ∫ ϑdx = 0.

Itsvimbo yakasungirirwa kumativi ese nechinguva panzvimbo yausingade uye dhayagiramu yemabatirwo akaitwa Ma naMb

Sl.11

** Chinyorwa chekupedzisira: ** kusarudzwa kwemuenzaniso we torsion kunoenderana nechikamu chakavhurika kana chakavharwa, kutetepa, nzira yekutsigira, nzvimbo yenguva yekuisa uye mukana wekupeta mahara. Kune chinhu chaicho, iyo inotakura-mutoro uye sevhisi muganho inotaura, ruzivo rwemunharaunda, majoini, kuneta uye kugadzikana zvinofanirwa kuongororwa, kwete imwe chete yakasarudzika fomula kubva kuchinyorwa.