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12341234123884()dVmFdt=dHMdt=M()()()m(t)m()dVmtFdt=dHMdt=111oxyz333oxyzAxyz222oxyz()AxyzAAxAz2.1AAyo()VV2.2222oxyzo2ox333oxyz3ox2oy2ox2ozV()333oxyzOV3ox3oy3ox3oz33xoy(l.1)RXYZ333oxyz()111oxyzO1ox1oy1ox1oz11xoy(l.l)111oxyz()AoAxyz111oxyzAxyzψϑγ2.3(a)ψ1oxAxAxψAyϑ1oxϑ'Azγ1oy1ox1ox1oyγ1OxAxyz111oxyz(),,LψϑγAxyzAyψAxyz′′(2.3b)Axyz,,xyz(T)xyzAxyz′′()yxxyLyzzψ′⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟′⎝⎠⎝⎠()cos0sin010sin0cosyLψψψψψ−⎛⎞⎜⎟=⎜⎜⎟⎝⎠⎟AyψAxyz′′Az′ϑ1Axyz′′(2.3c)()1zxxyLyzzϑ′⎛⎞⎛⎞⎜⎟⎜⎟′=⎜⎟⎜⎟⎜⎟⎜⎟′′⎝⎠⎝⎠()cossin0sincos000zLϑϑϑϑϑ⎛⎞⎜⎟=−⎜⎜⎟⎝⎠1⎟1Axyz′′1Axγ(2.3d)111oxyz()1111xxxyLyzzγ⎛⎞⎛⎞⎜⎟⎜⎟′=⎜⎟⎜⎟⎜⎟⎜⎟′⎝⎠⎝⎠()1000cossin0sincosxLγγγγγ⎛⎞⎜⎟=⎜⎜⎟−⎝⎠⎟Axyz,,xyz111oxyz()()()111xzyxxyLLLyzzγϑψ⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()()()(),,xzyLLLLψϑγγϑψ=()111,,xxyLyzzψϑγ⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠(),,Lψϑγ(22)(24)(26)(28)()coscossincossin,,sincoscossinsincoscossinsincoscossinsincossinsincoscossinsinsinsincoscosLϑψϑϑψψϑγϑψγψγϑγϑψγψγϑψγψγϑγϑψγψγ−⎛⎞⎜⎟=−++⎜⎟⎜⎟+−−+⎝⎠111,,xyz()1111,,xxyLyzzψϑγ−⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()(1,,,,TLL)ψϑγψϑγ−=31311,0,ikjkijkkikjijkijijllllijijδδδδ==⎫=⎪⎪⎪⎪=⎬⎪⎪==⎪=≠⎪⎭∑∑ABCABBCABLBCLACACABBCLLL=()2.4Axyz222oxyzθV2oxxAzVθνψVxAzox′AxAyoxAx′νψAyνψ2Axyz′()cos0sin010sin0cosyLνννννψψψψψ−⎛⎞⎜⎟=⎜⎜⎟⎝⎠⎟2Axyz′2Azθ()cossin0sincos000zLθθθθθ⎛⎞⎜⎟=−⎜⎜⎟⎝⎠1⎟()()(),coscossincossinsincoscossinsinsin0coszyvLLLνννννννψθθψθψθθψθψθθψψψ=−⎛⎞⎜⎟=−⎜⎟⎜⎟⎝⎠Axyz,,xyz222oxyz222,,xyz()222,xxyLyzzνψθ⎛⎞⎛⎜⎟⎜=⎜⎟⎜⎜⎟⎜⎝⎠⎝⎞⎟⎟⎟⎠3()V1oxV()111oxyz33oxyzαβαβ333oxyz3oyβ(2.5)31oxyz′()cos0sin010sin0cosyLβββββ−⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠31oxyz′1ozα111oxyz()cossin0sincos000zLααααα⎛⎞⎜⎟=−⎜⎟⎜⎟⎝⎠1nsin333oxyz111oxyz()()()coscossincossin,sincoscossisin0coszyLLLαβααββααβαβααβββ−⎛⎞⎜⎟==−⎜⎟⎜⎟⎝⎠333,,xyz()13113,3xxyLyzzβα⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()V2ox3oxνγνγ11xoyVoyoy3oy2oy3oy23νγ(2.6)2oxνγ()()1000cossin0sincosxLLννννννγγγγγγ⎛⎞⎜⎟==⎜⎟⎜⎟−⎝⎠()323322xxyLyzzνγ⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()(216)218()()333,vxxyLLyzzνγψθ⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()()(),,,vvLLLννψθγγψθ=()(218)217()()12112,v2xxyLLyzzβαγ⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()()(),,,vvLLLγβαβαγ=()dVVVdttδδ=+Ω×dVdtVtδδVVmVtδδ⎛⎞F+Ω×=⎜⎝⎠⎟222222222222222222xyzzyxyzxxzyzzxyyxdVVVdtFdVmVVdtFFdVVVdt⎛⎞+Ω−Ω⎜⎟⎜⎟⎛⎞⎜⎟⎜⎟+Ω−Ω=⎜⎜⎜⎟⎜⎟⎝⎠⎜⎟+Ω−Ω⎜⎟⎝⎠⎟⎟⎞⎟⎟⎟⎠VV222oxyz2ox22200xyzVVVV⎛⎞⎛⎜⎟⎜=⎜⎟⎜⎜⎟⎜⎝⎠⎝22,νψθ&&222oxyz()22200sin,0c0xyzLννννosψθψθψψθθθΩ⎛⎞⎛⎞⎛⎞⎛⎜⎟⎜⎟⎜⎟⎜Ω=+=⎜⎟⎜⎟⎜⎟⎜⎜⎟⎜⎟⎜⎟⎜Ω⎝⎠⎝⎠⎝⎠⎝&&&&&⎞⎟⎟⎟⎠(223)(224)(222)222cosxyzvdVmdtFdmVFdtFdmVdtθψθ⎛⎞⎜⎟⎛⎞⎜⎟⎜⎟⎜⎟=⎜⎜⎟⎜⎟⎜⎟⎝⎠⎜⎟−⎜⎟⎝⎠⎟dVdt2oxVddtθ()2oycos()vVddtθψ−2oz2.7VddtθtAdtdsBdθ22yaVρ=dsdsdtVdddtddtρθθVθθ====&22yVdaVdtθVθρ===&2cos()zaVddvtθψ=−νψ22πθπ−ddνtψ(225)1.RXYZ(220)24333oxyz222oxyz()()2122cossinsincosxTyzRXXXRLYLYYZRZZYZνννννιγγγγγγ−−−−⎛⎞⎛⎞⎛⎞⎛⎜⎟⎜⎟⎜⎟⎜===−⎜⎟⎜⎟⎜⎟⎜⎜⎟⎜⎟⎜⎟⎜+⎝⎠⎝⎠⎝⎠⎝⎞⎟⎟⎟⎠2.PoxP(217)P(218)PP1222oxyz333oxyz111oxyz1xP1yP1zP11100xyzPPPP⎛⎞⎛⎜⎟⎜=⎜⎟⎜⎜⎟⎜⎝⎠⎝⎞⎟⎟⎟⎠(),TLβα()313113,xxTyyzzPPPLPPPβα⎛⎞⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠()()()232323,xxTTTyyzzPPPLPLLPPPννγγβα⎛⎞⎛⎞⎛⎜⎟⎜⎟⎜==⎜⎟⎜⎟⎜⎜⎟⎜⎟⎜⎝⎠⎝⎠⎝111xyzPP⎞⎟⎟⎟⎠νg⎞⎟⎟⎟⎠cos(228)(227)()()222coscossincoscossinsinsinsincossincosxyzPPPPPPννναβαγαβγαγαβγ⎛⎞⎛⎞⎜⎟⎜⎟=+⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−⎝⎠⎝⎠3.G(143)Aymg0000AxAyAzGGGmG⎛⎞⎛⎞⎛⎜⎟⎜⎟⎜=−=−⎜⎟⎜⎟⎜⎜⎟⎜⎟⎜⎝⎠⎝⎠⎝G(216)22()222sin,0xAxyAyAzzGGmgGLGmgGGνθψθ⎛⎞−⎛⎞⎛⎜⎟⎜⎟⎜==−⎜⎟⎜⎟⎜⎜⎟⎜⎟⎜⎜⎟⎝⎠⎝⎝⎠θ⎞⎟⎟⎟⎠(226)(229)(230)(225)()()coscossinsincoscossinsincossincossinsincossincossincoscosvdVmdtPXmgdmVPYZmgdtPYdmVdtννννννναβθθZναγαβγγγαγαβγγγψθ⎛⎞⎜⎟−−⎛⎞⎜⎟⎜⎟⎜⎟=++−−⎜⎟⎜⎟⎜⎟⎜⎟−++⎝⎠⎜⎟−⎜⎟⎝⎠θ()111oxyzωdHHHMdttδωδ=+×=dHdtHdtδ1i1j1k1xω1yω1zωωHJω=⋅11111111111111xxyzxyyyzzxyzzJJJJJJJJJJ1x⎡⎤−−⎢⎥=−−⎢⎥⎢⎥−−⎣⎦1xJ1yJ1zJ11xyJ11yzJ11zxJH1ox1oy1oz1111111111000000xxxxxyyyyzzzHJJHJJHJωωωωωω⎛⎞⎡⎤⎛⎞⎛⎜⎟⎜⎟⎜⎢⎥==⎜⎟⎜⎟⎜⎢⎥⎜⎟⎜⎟⎜⎢⎥⎝⎠⎣⎦⎝⎠⎝11111yzzJ⎞⎟⎟⎟⎠(l)()()()xxzyzyxyyxzxzyzzzyxyxdJJJdtMdJJJdtMdJJJdtωωωωωωωωω⎛⎞+−⎜⎟⎜⎟⎛⎞⎜⎟⎜⎟+−=⎜⎜⎜⎟⎜⎟⎝⎠⎜⎟+−⎜⎟⎝⎠M⎟⎟xMyMzM111oxyz1ox11xoy0yzzxJJ==()()()()22yxxxyzyzyxyxzxyxyxyxzxzxyzyyzzzyxyxxyyxddJJJJJdtdtMddJJJJJMdtdtMdJJJJdtωωωωωωωωωωωωωωωωω⎛⎞−+−+⎜⎟⎜⎟⎛⎞⎜⎟⎜⎟−+−−=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎜⎟+−+−⎜⎟⎝⎠()xyzdxdtVdyVdtVdzdt⎛⎞⎜⎟⎛⎞⎜⎟⎜⎟⎜=⎜⎟⎜⎜⎟⎜⎟⎝⎠⎜⎟⎜⎟⎝⎠⎟⎟VV222oxyz2ox()()222cos,,0sin0cossinxxTTyyzzVVVVLVLVVVVνννθψθψθθθψ⎛⎞⎛⎞⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟===⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−⎝⎠⎝⎠⎝⎠⎝⎠cossincossindxdtVdyVdtVdzdtνθθθψ⎡⎤⎢⎥⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥−⎢⎥⎣⎦⎢⎥⎢⎥⎣⎦Axyz,,xyz()ψϑγ1xω1yω1zωω(2.3)()()()00TxzLLγθψ&()()00TxLγϑ&()00Tγ&()()()111000000sin1sin0coscossin0coscossincossincos0cossincosxyxzxzLLLωγωγϑψγϑωψϑγϑγψϑγϑγϑγγψψϑγϑγϑγγϑ⎛⎞⎛⎞⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟=++⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠+⎛⎞⎡⎜⎟⎢⎥=+=⎜⎟⎢⎥⎜⎟⎢⎥−+−⎝⎠⎣&&&&&&&&&&&&⎤⎛⎞⎜⎟⎜⎟⎜⎟⎦⎝⎠1111cossincossin0coscos0sincosxyztgtgϑγϑγωγγγψωϑϑϑωγγ−⎡⎤⎛⎞⎛⎞⎢⎥⎜⎟⎜⎟⎢⎥=−⎜⎟⎜⎟⎢⎥⎜⎟⎜⎟⎜⎟⎢⎥⎝⎠⎝⎠⎣⎦&&&(()sincos1cossincoscossinyzyzxyzddtddtdtgdtϑωγωγψωγωγϑγ)ωϑωγωγ⎛⎞⎛⎞⎜⎟+⎜⎟⎜⎟⎜⎟⎜⎟=−⎜⎜⎟⎜⎟⎜⎟⎜⎟−−⎜⎟⎝⎠⎜⎟⎝⎠⎟90ϑ=oψ[29]()sdmmtdt=−dmdt()smt()()smtsm(238)()00ftstmmmtdt=−∫0m0tft22θνψνγϑψγαβ(2.8)(231)(237)θνψϑψγαβνγϕ(2.9)121coscoscoscoscos212coscosϕααβ=+βγγ+ijkAAxyz12ll⋅(239)12121212lllililjljlklk⋅=⋅⋅+⋅⋅+⋅⋅oxlOA212226(240)1oz212lAxyz222oxyz111oxyz()()sincoscossinsinsincossincossinννβθγψψϑγψψθϑ=⎡−+−⎤−⎣⎦γoxoxl1212l()coscoscoscossinsin/cosναϑθψψϑθ=⎡−+⎤⎣β⎦ozozl1212l()()coscoscossinsinsin/cosvννγγψψϑγψψ=⎡−−−⎤⎣⎦β(241)(243)αϑθ=−vγγ=(0α≈)vβψψ=−ϑ*ϑϑϑϑ∗∆=−ϑ∆zδ()zKKϑϑδϑϑϑ∗∆=−=∆Kϑiεix∗ixiiixxε∗=−i=1,2,3,4iεiε()()()()1234xyzpffffδεδεδεδε⎫=⎪=⎪⎬=⎪⎪=⎭*0iiixxε=−=1*2*34*0000VVεθθεψψεγε=−=⎫⎪=−=⎪⎬==⎪⎪=−=⎭12*34*0000VVεθεγγεβε==⎫⎪=−=⎪⎬==⎪⎪=−=⎭(247)(248)θ∗νψ∗γ