A. N. BIRBRAER. J. ROLEDEREXTREMEACTIONSON STRUCTURESSaint PetersburgPublishing House of the Politechnical University2009. . . . - 2009 624.04 38.112 64 , , . . . . /. .,. . ..:-.-,2009.594 .-.--.- , ,,,,.--, . .--. - .. 72. . 269. . : 143 .Bi rbraerA. N. ExtremeActionsonStructures/A. N. Birbraer,A. J. Roleder.St.Petrsburg.:PublishingHouseofthePolitechnicalUniver-sity, 2009. 594 p.The book deals with problems of designing structures intended to withstandextremeactions.Techniquesfordynamicanalysisofstructuressubjectedtoshort-termloadsinelasticandinelasticdeformationstagesisdescribed.Thebookpresentsmethodsfordeterminingloadscausedbyextremeactions,i.e.impactsfrommissilesofvariousnature, explosions,tournedosandhurricanes,loaddrop,structurecollapse.Principlesofconsideringextremeactionsinde-sign of nuclear power plants, including probabilistic assessment of their hazardandrisk,aredefined.Thebookcontainsnumerousexamplesofcalculations.Table of contents, which provide a deeper inside in the book contents, is givenat the end of the book.The book is for use by design engineers and scientists. It may also be usedby civil engineering students and postgraduates. . ., . ., 2009 - ISBN 978-5-7422-2370-2 , 2009............... ............... ............ ... 13 .......... ............... ............. .. 16I 1. .............. .......... . ..171.1. ....... 171.1.1. ............ .... 171.1.2. ... ... ... ..... 181.2. ....................211.2.1. ............... 211.2.2. ............... ............ 231.2.3. - .............. ............. 30 2. .................... 332.1. ( ) 332.1.1. ................... .......... 332.1.2. ................... ....... 352.2. ............... 372.2.1. ................. 372.2.2. ............. 382.2.3. ............... 382.2.4. ................... ................ 412.2.5. - ................... ......... 432.3. ... ... ... ... ... ... ... ... ... ..... ... ... ... . 472.3.1. . . ... ... ... . 472.3.2. .. ... ... ... .. .. .. . 482.3.3. .. ... ... .. 492.4. - .................................532.4.1. ............. 532.4.2. - ........................................... 542.4.3. ................ 552.5. .... 55 3. ................. 573.1. .......................... 573.1.1. ............................. 5763.1.2. - ................................. 663.1.3. ........................... .........703.1.4. ..... .... 723.1.5. 733.2. .... .... ... 773.2.1. .... .... .... .... .... .773.2.2. .... .... .... .... .... .... .. 773.2.3. - .... .... .... .... .... .... .... .... .... .793.2.4. ........ ....................... 803.2.5. 823.3. - ......... ... ... ... ... ... ... ... ... ... ... 823.3.1. - ..................................... 823.3.2. ... ... ... ... ... ... ... ..... 833.3.3. ......... ... ... ... ... ... ... ... ... ..... 843.3.4. 50 ........... 843.3.5. ... ... ... ... ... ... ... ... 853.3.6. ... ... ..... 853.4. ... ... ..... 86 4. ...... 884.1. ....................................... 884.1.1. ... ... ... ... ... ... ... .... 884.1.2. .....894.1.3. - ... ... ... ... ... ... 954.1.4. ... ... ... .... 984.1.5. ............ .... 1014.2. . .... 1044.2.1. ... ... ... ... ... ..... 1044.2.2. ................... 1044.2.3. - ................ .......... 1084.2.4. n ................ ............ 1084.3. ... 1104.3.1. ... ... ... ... ... ... ... ... ... ... 1104.3.2. - .... . 1124.3.3. ... ... ... ... ... ... 1134.3.4. () .. ... .. 11674.3.5. -..................................... 1184.3.6. .. ... .............. 1284.4. .. 129 5. 1305.1. ... ... . 1305.2. ... ... ... ... ... ..... 1315.3. ............................ 1325.3.1. ............ 1325.3.2. ... 1365.3.3. ... ..... ..... 1385.3.4. ... ... ... .....1405.4. ................1425.4.1. - ... ... ... ... ... ... ... ... ... ... 1425.4.2. ...... .. 1435.4.3. - .......... 1495.4.4. ................ 1505.4.5. - .................. 1515.4.6. - ..............................1545.5. - ..............................1545.5.1. ...................... 1545.5.2. ........................ 1555.5.3. ... ... ... ... ... 1595.5.4. ... ... ... ... ... ... ... .... 1625.5.5. .. ... ... ... ... ... ... .... 1635.5.6. - ... ... ... ... ... ..... 1685.5.7. ... ... ... ... ... ... ... ... 1685.5.8. - ... ... ... ... ... ... ... ....... 1685.6. ..................... .....169 6. - ... ... ... ... ....1706.1. 1706.2. .... 1736.3. -......... ... ... ... ... ... ... ... ... .....1746.3.1. ... ... ... ... ... ... ... . 17486.3.2. ... ... ... ... ... ... ... .... 1746.3.3. ... ... ... ... ... ... .. 1776.3.4. ... ... ... ... ... ... ... .... 1816.3.5. .................... 1856.4. ... ... ... ... ... ... ... ..... 187I I 7. ... ... .... 1887.1. 1897.2. - - ....................................... 1897.3. ...... 1947.3.1. ... ... ... ..... 1947.3.2. - ....................................1957.3.3. - ................................2107.3.4. ................................ 2127.4. ... ... ....214 8. ...................2168.1. ...................2168.1.1. ...................... 2168.1.2. ....... 2198.1.3. ... 2228.1.4. 2248.2. ........................ . 2258.2.1. ...................... 2268.2.2. ... ..... 2308.2.3. - ...... ... ... ... ... ... ... ... ... ... ... 2368.2.4. ............... . 2388.3. ...... 2398.3.1. ... ... ... ... ... ... .. 2398.3.2. .............. 2408.3.3. ............................ 2438.3.4. .................. 2498.3.5. ... ... .....2548.4. .......................... 2568.4.1. ................ 2568.4.2. ......................... 2568.4.3. ... ... ... ... ... ... ... ... ... ... 2589 9. ............... 2609.1. ............................ 2619.1.1. ................................ 2619.1.2. - - 2629.2. ............................. 2669.2.1. .... 2669.2.2. . ... ... ... ... . 2689.2.3. - .......... 2749.2.4. - .. ... ... ... ... ... ... ... ... ... 2819.2.4. - - ......... ... ... ... ... ... ... ... ... ... ... 2819.3. ........................... 2879.3.1. ... 2879.3.2. - ............................... 2959.4. .................... 2999.4.1. ..... 2999.4.2. .... .............. 308 10. ....................30910.1. ...30910.1.1. 30910.1.2. .......................... 31810.2. .................................... 31810.2.1. ............................. 31910.2.2. ... 32210.2.3. .................... 32510.2.4. ....................... 32810.3. ....... 331 11. - ......................................... 33411.1. .................................... 33411.1.1. ...................... 33411.1.2. - ......................................... 33411.1.3. ..................................... 34711.1.4. 35211.1.5. - ................... 35311.2. ................. 35511.2.1. .............................. 35511.2.2. ..................... . 3561011.2.3. ........................... 35811.2.4. .............. 35911.2.5. ................ 36011.2.6. ..........................36011.2.7. . ................. 36111.3. ......... 367 12. ................. 37012.1. ..... 37012.2. ......... 37312.2.1. ................................. 37312.2.2. ......................... 37612.2.3. ............ 37712.2.4. ...... 38012.2.5. - SCAD Office................... 38812.3. . ................... 38912.3.1. ................................. 38912.3.2. ........ 39012.3.3. ................. 40712.3.4. ........................... 41212.3.5. .............. 417 13. ......... 42313.1. ................. 42313.2. .......................................... 42313.2.1. ....... 42313.2.2. ........ 42813.2.3. ....................... 42913.2.4. .............. 43113.3. .......................................... 439I I I 14. - ............................ 44214.1. ..................................44214.2. .................44514.3. - .................................. 44814.4. -............................................ 45314.5. .................... 45714.6. ................................... 4591114.7. ......... 46014.8. ................. 46114.9. ........................... 46214.10. ............................. 46314.11. - ......................................... 46414.11.1. ...................... 46414.11.2. ........... 46414.11.3. ................ 466 15. ................................... 47115.1. ..................................47115.1.1. ....................... 47115.1.2. .............................. 47215.1.3. ....................... 47215.1.4. ...... 47415.1.5. ............ 48115.1.6. ....................................... 48415.2. ................................. 49415.3. ............................49715.3.1. ........ 49715.3.2. ..................... 49915.3.3. ....... 506 16. .............................. 50916.1. ..........................50916.2. .........51016.2.1. ......... 51016.2.2. ...... 51016.2.3. .............51116.3. ................................... 51316.4. - .............................................52216.5. ......... 524 17. ...................... 52617.1. ..........................52617.2. ..........................52717.2.1. ....... 52717.2.2. ...................................... 52817.3. ...........5381217.4. ...........54417.5. 546 18. ......................... 55118.1. 55118.1.1. ................... 55118.1.2. 55218.1.3. , , ..................................... 55418.1.4. - ....................................... 56018.2. ..... 56618.2.1. ....................... 56618.2.2. .............. 56718.2.3. 56718.2.4. ..................................... 56918.2.5. ....................................... 57018.3. ...........................571 ................................ 573 .................................... 581Contents ................................................ 587-,, , .. -. , -,,.,,,,,- .,,, (). -. , , - .,- , .,,(),,,,,..- : -,- , . - , , -- - , - .,- , . ,-(- , , [12]). -,,,- , . , . .-, , -14.:- , -(..-),..- , -, - .-.;;,-;,-;,;.:, , ,,-- . -,,,,,, .., .,--.,- , -.-, .I-,,.,. , . .,,--,- .II,.-,,,,.III- . , .,15 : .;- ; .-- . -.-,,- .- : -, ..,- - , , ( -).--,-. -.,-,.,- , .-,,,-( ).-, . .--- . 8.1, 8.2, 11.1 16.4 . . , 2.3, 3.1 4.3 . . .,-,,. . , . , .16 - ( , ) - - - () CPV Conditional Probability Value ( , )CQCComplete Quadratic CombinationDBPVDesign Basis Probability Value ( , )MSK-64 , 1964 .SDV Screening Distance Value ( , )SPLScreening Probability Level ( , )SRSSSquare Root of the Sum of the Squares ( , )ZPAZero Period Acceleration ( , )I 1 ( -). , (,,),,.,-- , , . .1.1. 1.1.1. -,.1985 .[3], [4], [5], -.,- , , - [29], - , - [3].-.,M 150,M 200..-, 12.5, 15 . [77]) . ( . V 64 1 1 098 0BM= ,(1.1)V; , , V = 0.135, V = 0.18..1. 18.[3] -I, A-III ., A-240, A-400 .. . 1.1[3],..,,,[29], ., . 1.1. A-I A 240 A-III A 400 A-V > B 500A-II A 300 A-IV > B 500 Bp-I B 500 . -. , ,- [3] R. -,fc. 07 1 R fc. = . (1.2),-[5, 6]-Rb.,- R, Rs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sPQ. 1.4. - - ,.. . - .,,,-.-,. 1.4 Qs/2, . II ,-, .(,..)[29, 50]I-, .I- . - ,-().- . - .1.2. 23I . -- ., I (.. -),.( I). - I, - R . .1.2.2. - [29]. [50] , - . .,I,- . -.-[29].--[4,6],-(..-). I, -QQY0 Y Y 0Y. 1.5. ( ) . - Y Q ,. 1.5().0 Y Y0, Y Y0 . [29]. .--.1. 24(..-[4, 6])-K.-Y,-YY0,-, Y0:0 00YY YYYK+= = . (1.5),-I,K = 3YiY.,,I,K = 1 Yi Y0. -:) ,- R ,0200003 0hSlMMERY||.|

\|+ =.; (1.6)) , ,( )20075 . 0'003 . 0SlRRhY = ; (1.7)) , ,20.75 0 003 0SlF R F R Nb RY +=. ,. (1.8)-,----,I, -00.20.1 0.20.1x h0. 1.6. 1.2. 25,.- i i , (1.9)< +=. . ., ...02 0 2 002 0 003 0035 0i(). (1.10) . 1.6., .. (1.9)(1.10), - b = 1 . (1.5)(1.10) :R ,.R - (), ; R ,; , ;, (-) ) (0 bh F = ; ) (0 bh F= ; (1.11)F ,F () , 2;b , ;h0 , ;l0 , ;a - , ;N , ;M , - R ,( ) ( ) ' 5 . 0 5 . 0 . 0a x R F x h R F M + = ;(1.12)M , R = ;- 205 0 R bh M . = ;(1.13)x - ) ( bR R F x = ;(1.14).1. 26 ;- 0 h x = R R = , (1.15) ) ( 0R bh N R R + = ;(1.16)R ,-- :||.|

\|+=1.1110 4 0190.RR; (1.17)0 ,9 010 8 85 0 R = . .(1.18)I- 0 , , , - , 10 %;S ,-.-S . 1.3. nn nM M MM S M S M SS+ + ++ + +=......2 12 2 1 1,(1.19)S1M1,S2M2,,SnMn-SM.20Sl f = 1 -M,-, . 1.3. S [29]ql41= SPl31= S1.2. 27 . 1.3lPal|.|

\| =6 21 aa S0.5lP0.5l121= SalPlalP6 812aS =lq485= SPlPal) () (aa aS+ + =1 62 320.5lP0.5lalPalP) () (aa aS4 1 124 3 2 12+ +=al0.5l 0.5lP Pqal P) () (K aK a aS+ ++ + +=8 2 485 4 3 16 82, PqlK =qPl) ( KKS++=2 123 8,PqlK =0.5l 0.5lqP) ( KKS++=2 485 8, PqlK =Pallq) () (K aK a aS+ + =2 123 3 42, PqlK =P P alqall) () (K aK a aS+ + =8 485 4 3 162, PqlK =qPalPl) () (K aK a aS+ ++ +=2 2 123 3 4 82,PqlK = - . . 1.4 -.1. 28 , [127, 135]. 1.4. K, [127, 135] KI. 1. :) ) 4 (0 Y l ) ) 4 (0Y l 2. ( ) 1.33. ( ):) 1.0) , 1.3) 3.0II. 4.(-):) (U, V, T- ..) 12.5) (, ..) 205. :) < 10 5) 10 16. 57. 2y =. 1.4(- [29] - ):Y0 [ , (1.5)];l ; , [80]; ;u ,- ; ,07 0 0065 0. . =cd , (1.20) d - ; c -1.2. 29( c ); .,. 1.4, [135]:1. -,.-, , K -.2. ,. 1.4, :bdAdtRRssb 24 . 1;sb s sRRbdA A25 0. , (1.21)Rb;Rs; As ; sA -;t;b ; d - .3. (1.21),,- (1.20). , -,,-150500 . , (1.20) .4. K = 20,-. 1.4,:26.4.-,-,,-.- [80].5. -,,,-.. 1.4- 2.6. .1. 30,. 1.4, .- - 20 %.1.2.3. - -.-. , - .-(.5).,,,,- , -.- . -,-, -.-, [61]. , , - , -:K,. -[29] [127, 135] (. 1.4). [29] - K , -- . , . 1.4 -[,- . (1.20)].K-1.2. 31,: K , --. , -Y0,.(-), Y0 , ---- .. 1.7-[127]--,.Y = 6.7 --.-- Q = 25.1 Y, 0 2 41020Q, 306QI2I3I1Y. 1.7. ( -.. 4).Y0- : I1 = I - ( ); I2 = I -,. 3.4;I3 = (I1+I2)/2. , , I1, - K = 61.9, I2 34.5, I3 7.7, .. - . , .. , 5 %. ( ). , - . , , -:, : (. . 5.5).-,. -.1. 32 Y0, (. . 5.22), - K . , K , -, . -,,- .[29]-,,.,,-.2 -.- , ,. .(),.-,-(, ,). .,- . - . - : -,-- . -. .,-, .2.1. ( )2.1.1. -(),. 2.1.- . 4.1.,-(,*) 0 = + + kx x b x m & & & , (2.1)

* , , , , , , .. 2. 34 m , ; b , /(/); k ,/;x& &, x& x , .xkmyxb. 2.1. - m(2.1)- 0 22= + + x x x & & & ,(2.2) -,mk= , /;(2.3) ,kmbmb22= =.(2.4).,mg.- x x k mg = . (2.5)k(2.3),- xg= , /. (2.6)( ) 00 x x = ) ( ;00 x x & & = ) ( ,(2.7) < 1, ) sin( + =t Ae xDt, (2.8) D ,21 =D; (2.9)A ,220 0 20Dx xx A ) ( ++ =&; (2.10) ,Dxx x00 0 +=&. (2.11)2.1. ( )35 .- f, (), T, ( ). - 21= =Tf . (2.12)(2.9),D = 1. = 1 . ,-,.(2.2)(). , .. ==. || || } ]{ [ } {;} ]{ [ } {j i Mj iMi iTijTi 0 02 (2.36) (2.34), := >=. || ||;} ]{ [ } {j ij iKi ijTi 0 02 2 (2.37),, .||i||2 = 1(-}~{i ). :} ]{ [ } {} {|| ||} {}~{iTiiiiiM = = . (2.38) :==.;}~]{ [ }~{j ij iMiTi 1 0 (2.39)2.2. 43==.;}~]{ [ }~{j ij iKiiTi 02 (2.40)2.2.5. ()(2.25)., . , (2.24).-. 2.3.2.-, . ,-,-(2.36)(2.37).,(2.24)-,. , (, ).- . .,(2.24) = == =njj jnjjt q t x t x1 1) ( } { )} ( { )} ( { , (2.41){xj(t)}j-(), - qj(t) *, -.(2.41).-(2.41)(2.24)- qj t, } { } { ] [ } { ] [ 01 1= + = =njj jnjj jq K q M & & . (2.42) . (2.41) i- {i}T:

* , qj(t) , , .. 2. 44} { } { } { ] [ } { } { ] [ } { 01 1Tinjj jTinjj jTiq K q M = + = =& & . (2.43)(2.36)(2.37),i-,, 02 2 2= +i i i i iq q || || || || & & . (2.44) (2.44) (2.1), , iq& & ,qi .- :i- ( ) } ]{ [ } { || ||iTi i iM m = =2; (2.45)i- 2 2|| ||i i ik = . (2.46) (2.44) ||i||2, 02= +i i iq q & & , (2.47)i-.-,.,(2.24)-,[12]., (2.47). 0 22= + +i i i i i iq q q & & & . (2.48) ..(2.48)-(2.2).- (2.25) - (2.24). {x0} } {0x& :==njj jq x10 0} { } { ;==njj jq x10 0& & } { } { . (2.49) jq0 jq0& - j- . - (2.49) {i}T[M]:2.2. 45. } ]{ [ } { } ]{ [ } {; } ]{ [ } { } ]{ [ } {====njj jTiTinjj jTiTiq M x Mq M x M10 010 0& & (2.50) (2.36) - i = j:i iTiq x M020|| || } ]{ [ } { = ;i iTiq x M020& & || || } ]{ [ } { = . (2.51) 200|| ||} ]{ [ } {iTiix Mq= ;200|| ||} ]{ [ } {iTiix Mq && = . (2.52)(2.48)- (2.8):) sin(i Diti it e A xi i + =, (2.53) Di i- ,21i i Di = ; (2.54)Ai i- ,220 020Dii i i ii iq qq A ) ( ++ =&; (2.55)i i- ,Di ii i i iiqq q 00 0 +=&. (2.56),.,,- .. (2.41):=+ =nii Diti it e A t xi i1) sin( } { )} ( { . (2.57),,- () -., (2.57) n, s < n . -, n . , -, .. 2. 46 0 0 0T U E + = ,(2.58) U0 ,} ]{ [ } {210 0 0x K x UT= ; (2.59)T0 ,} ]{ [ } {210 0 0x M x TT& &= . (2.60) , .. (2.57) i = 0, E0. ., (2.57):=+ =nii i i i it A t x1) cos( } { )} ( { & . (2.61) =+ = =nii i i i iTiTt A M t x M t x t T12 2 22121) ( cos } ]{ [ } { )} ( ]{ [ )} ( { ) ( & &. (2.62) (2.36)=+ =nii i i i it A t T12 2 2 221) ( cos || || ) ( . (2.63)s,-Tmax,s(2.63), E0 (, 5 %). . ,()..i-:} { }~{i i ik = . (2.64) 2 2 2|| || }~]{ [ }~{ ||~||j i iTi ik M = = . (2.65)(2.64)(2.65)(2.52),,- , ,ki.(2.55), :2.3. 47iiiAkA1=~. (2.66) i . (2.64) (2.66) (2.57) = + ==nii Diti it e A t xi i1) sin(~}~{ )} ( { =+ =nii Ditiii it e Akki i11) sin( } { . (2.67)ki.(2.57),,,..., , .. , , .(2.38).1,- .2.3. 2.3.1. -nR(t). ) ( } { } ]{ [ } ]{ [ t R J x K x M = +& &, (2.68) {J} :. ; ; ; 100 +=+ns kkaaJs kkMMMMMM) (} {(2.69) {J} , s -k, k+1, , k+s ak, ak+1, , ak+s,11= + + ++ + s k k ka a a K ..2. 48 , - .2.3.2. ,t.- .-.-., (., , . 8 9), - ., . 9.7, Phantom RF-4E. 0.07 , - -.-,-0.15 0.2 .1/10 1/12,- . . 9.7, -,80 1,..., 0.0125 , ( ) 001 . 0 0012 . 0 0125 . 0 12 / 1 10 / 1 t . t = 0.001 , , -150 200. - (-,MSC/NASTRAN[123125]).- t,,.-,-t (,.)., a [2, 37] .2.3. 492.3.3. . -.,,- (2.41). (2.68) ) ( } { ) ( } { ] [ ) ( } { ] [1 1t R J t q K t q Mnjj jnjj j= + = = & &. (2.70) (2.70) { i}T[M]:) ( } { } { ) ( } { ] [ } { ) ( } { ] [ } {1 1t R J t q K t q MTinjj jTinjj jTi = + = =& &. (2.71)(2.36)(2.37)- , i-:) ( } { } { ) ( || || ) ( || ||2 2 2t R J t q t qTi i i i i i = +& &. (2.72) ( ), ) (|| ||} { } {) ( ) ( t RJt q t qiTii i i22 = + & & . (2.73)- i- , i- qi(t). , - , - . ) (|| ||} { } {t RJq q qiTii i i i i i222 = + + & & & . (2.74).,.2.1.2,, . . 2|| ||} { } {iTiiJ= . (2.75)ii-.-,-.- (2.75) ) ( 22t R q q qi i i i i i i = + + & & &. (2.76)- i.2. 50R(t),(2.15).- . , i i iq = . (2.77)) (t Ri i i i i i= + + 22& & &. (2.78)(2.78)(2.41),- = =njj j jt t x1) ( } { )} ( { . (2.79) { i}i i-. } { } {j j j = . (2.80) ==njj jt t x1) ( } { )} ( { . (2.81).,-.,}~{j }~{j , (2.64):iiTiiTiijTijik kJkJ kJ = = = = 1|| ||} { } {|| ||} { } {||~||} { }~{~2 2 22. (2.82)(2.81)}~{j ,- (2.80):} { } { } {~}~{ }~{j j j j j j j jkk = = = =1. (2.83),{j},-, {x(t)} .-.-,(2.81)n,- r < n . {i}. (2.73) [M]{ i}:) ( } ]{ [ ) ( } ]{ [ ) ( } ]{ [2t R M t q M t q Mi i i i i i i = + & &. (2.84)2.3. 51 (2.36)(2.37) , } ]{ [ } { } ]{ [ } {jTi jTi iK M =2, (2.85), ,} ]{ [ } ]{ [j j iK M =2. (2.86) (2.84) ) ( } ]{ [ ) ( } ]{ [ ) ( } ]{ [ t R M t q K t q Mi i i i i i = + & & . (2.87) (2.87) n : = = == +niinii inii it R M t q K t q M1 1 1} { ) ( ] [ ) ( } { ] [ ) ( } { ] [ & & . (2.88) (2.70) (2.88), , , , :) ( } { } { ) ( ] [1t R J t R Mnii == .(2.89) R(t) 0, , } { } { ] [ J Mnii ==1 .(2.90) {i}:} { ] [ } { J Mnii11== .(2.91) , . -k()R(t),- - . r+1k(),-,.- - S(). . 4.-. , (2.81) r < n r. {i} = + = =riinr iiJ1 1} { } { } { . (2.92) .2. 52+ ==nr ii1} { }~{ . (2.93) r+1., :k() k( r+1) const > r+1. (2.94)- , - :) ( }~{ ) ( } { )} ( { ) ( } {1 1 1+= =+ < < rrjj jrjj jk t t x t .(2.95), : , . ) ( }~{1 += rk (2.96) .(2.91),sk,k+1, ,k+s, mk, , mk+s. {J}(2.69).- (2.75), 2|| ||,is k i s k ik kia a + ++ += K.(2.97)} { } { ] [) ( 1 sJ J M =,(2.98) {J(s)} , k- (k+s)- - ak/mk, , ak+s/mk+s, : 100 +=+ +ns kkm am aJs k s kk ksMMMMMM) (} {) ((2.99)2.4. 53 (2.91) } { } {|| ||) ( , sniiis k i s k ik kJa a=+ +=+ +12 K.(2.100),,k-,mk,(2.100)-:} { } {|| ||) (112Jniiiik==, (2.101) {J(1)} : 10101=nk m JkMMMM/ } {) ((2.102),. , (2.75)} { }~{ JTi i = . (2.103) , (2.100) } { }~){~ ~() (,snii s k i s k ik kJ a a = + +=+ +1 K , (2.104) (2.101)} { }~{~) (11Jnii ik== . (2.105)2.4. 2.4.1. - ( ). - ().-,-,-. , - (-,),,-. 2. 54(,).-,..-,[12]. ., .()) (t X& &.,[M],[K].,,) ( } ]{ [ ) (0t X J M t R& & = , (2.106) {J} , (,-,0).(2.106)(2.68), :) ( } ]{ [ } ]{ [ } ]{ [0t X J M x K x M& && & = + . (2.107)2.4.2. , , (2.41). - (2.107) - i- {i}T:) ( } ]{ [ } { ) ( } { ] [ } { ) ( } { ] [ } { t X J M t q K t q MTinjj jTinjj jTi 01 1& && & = + = =. (2.108)(2.36)(2.37), i = j, . ) ( } ]{ [ } { ) ( || || ) ( || || t X J M t q t qTj j j j j j 02 2 2& && & = + . (2.109), q j(t):) (|| ||} ]{ [ } {) ( ) ( t XJ Mt q t qjTjj j j 022& && & = + . (2.110)- (2.20) , j- . 2|| ||} ]{ [ } {jTjjJ M= . (2.111)2.5. 55, . j = 0, - . (2.77)(2.81). (2.77)(2.80) ==njj jt t x1) ( } { )} ( { . (2.112)2.4.3. { j} } { } {1Jnjj == .(2.113) - ( - . 4). .[22],{ j}n--,.. . {J}:==njj ja J1} { } { ,(2.114) aj , . , (2.114) - i- :==njj jTiTia M J M1} { ] [ } { } ]{ [ } { . (2.115)(2.36)0, i = j:2|| || } ]{ [ } { } ]{ [ } {i i iTi iTia M a J M = = . (2.116) 2|| ||} ]{ [ } {iTiiJ Ma= , (2.117)..aii. (2.116), (2.113) (2.115).2.5. - , -.,,-.2. 56 , - . [], . (2.36)- :[] [ ][] [ ]2|| ||iTM = , (2.118)[ ]2|| ||i , i- - i- .(-,- ):[] [ ] [] ( ) [ ]1211 1 = || ||iTM . (2.119) (2.119) [], []T,[ ] [][ ] []TiM =121|| || . (2.120)[ ]12 || ||i , , .- [] = [K]1. (2.37) :[] [][] [ ][ ]2 2|| ||i iTK = , (2.121)[ ]2i , i- i- . - [], []T, [ ] [] [][ ] [ ] []Ti iK = = 12121 || || , (2.122)[ ]12 i , . , [ ] [ ] Ei=2|| || , (2.123)[E].(2.120)(2.122) [ ] [][]TM =1; (2.124)[ ] [] [][ ] []TiK = = 121 . (2.125) 3 , .. , .,,,,-,,.- .,.,,- . -.,( ) ,- (,-,.).,-, . - [53, 54, 63,85].3.1. 3.1.1. .y (x, t)-xt(. 3.1).- 0 ) ( ) ( ) (222222=+||.|

\|tyxxyx I x Ex , (3.1)E(x),;I(x) , 4; (x) , /.l-, 02244=+tyxyEI . (3.2) (3.1) (3.2) . .. 3. 58y(x,t)xxyl. 3.1. -. .-, ,,x = 0x = l.- x = 0: , ..0 ) 0 ( = y0) 0 (=xy; (3.3)-, 0 ) 0 ( = y0) 0 (22=xy; (3.4) -, 0) 0 (22=xy 0) 0 (33=xy. (3.5) : - , . [22]. .. t = 0:) ( ) 0 , (0x y x y = ; ) () 0 , (0x ytx y&=. (3.6).,, ) ( ) ( ) , ( t q x t x y = , (3.7).. , -x,t.-,..(3.2).(3.7)-, ) () () () (xx EIt qt qIV =& &.(3.8)t,x.,3.1. 59 , .. -. 2,(x)q(t) :0 ) ( ) (2= xEIxIV ; (3.9)0 ) ( ) (2= + t q t q & &. (3.10)..,,-,. ) ( ) ( ) ( ) ( ) (4 3 2 1kx V C kx U C kx T C kx S C x + + + = ,(3.11)42EIk= ; (3.12) .. :). sh (sin21); sh (sin21); ch (cos21); ch (cos21kx kx V kx kx Ukx kx T kx kx S = + = = + =(3.13)- . (3.11) - (3.4), (3.3) - (x):0 ) 0 ( = ; 0 ) 0 ( = ; 0 ) ( = l ; 0 ) ( = l . (3.14) (3.11) C1 = C3 = 0. - C2 C4:= += +. 0 ) ( ) (; 0 ) ( ) (4 24 2kl T C kl V Ckl V C kl T C(3.15), :0) ( ) () ( ) (=kl T kl Vkl V kl T. (3.16) , , - k:0 ) ( ) (2 2= kl V kl T . (3.17). 3. 60 (3.13) 0 sin sh = kl kl . (3.18),shkl = 0,(3.12), = 0, .. . - . , shkl 0, , ,0 sin = kl . (3.19) :) , 3 , 2 , 1 ( K = = n n l kn . (3.20) . n- - (3.15), :2 4) () (Cl k Tl k VCnn = . (3.21), (3.11) ) sh sin sh (sin) ( 2) (2x k l k l k x kl k TCxn n n nnn+ = . (3.22)(3.19),n- ( ) :x k C xn n nsin ) ( = , (3.23) Cn ,) ( 2sh2l k Tl kC Cnnn = . (3.24) C2 , Cn.,,n(x)-.,- . . [22].1. , .2. - . - :a) :

==lnm nm nm ndx x x02, || ||; 0) ( ) ( (3.25) || n||2 , ,3.1. 61

=ln ndx x02 2) ( || || . (3.26)( );) :== ln nm nm n km ndx x x02 2. || ||; 0) ( ) ( (3.27) :==lnm nm nm ndx x x x02, || ||; 0) ( ) ( ) ( (3.28)== ln nm nm n km ndx x x x I x E02 2, || ||; 0) ( ) ( ) ( ) ( (3.29)=ln ndx x x02 2) ( ) ( || || . (3.30)3. ( -).,f (x),- , 0 x l n(x): ) ( ) (1x a x fnn n == . (3.31).-(3.10).,,.- , . 0 ) ( ) ( 2 ) (2= + + t q t q t qn n n n n n & & &. (3.32)(2.2),-.k n,,- n(x) n, - (3.12) EIlnn22= ,/,(3.33) n = n,n = 1, 2, 3 . 3. 62 , ( ) .,,,- , .,,- , . 3. 1. - n(), lx= nDnn1 2 23 34 4xlsinnn 5n1 1.0178 4.7302 0.9992 7.8533 1.0000 10.9964 1.0000 14.137xl), cos ch (sh sin n n nn nD ++ n nn nnD cos chsin sh=n 52) 1 2 ( + n1 1.0000 3.9272 1.0000 7.0693 1.0000 10.2104 1.0000 13.352xl), sh (sincos ch n n nn nD ++ n nn nnD sin shcos ch++=n 54) 1 4 ( + n1 1.3622 1.8752 0.9817 4.6943 1.0008 7.8554 0.9999 10.996xl) cos ch (sh sin n n nn nD ++ n nn nnD cossin++=chshn 52) 1 2 ( n*) . n 3.1. 63 (3.32) (2.6)) sin( ) (n Dntn nt e A t qn n + =, (3.34)Ann-;n; Dn n- :21n n Dn = . (3.35) *)n) ( n ) ( n 1.273200.44240 nn] ) 1 ( 1 [ 2 n nsin2 n ncos30.815200.36370)] cos ch (sh sin [2 n n nn n nD + ++ )] sin sh (ch cos [3 n n nn n nD ++ 1.23040.11700.47280.0620)] sh sin (cos ch [2 n n nn n nD + +)] ch (cossin sh [3 n n nn n nD + 0.574850.441120.354540.1819)] cos ch (sh sin [2 n n nn n nD + ++ )] sin sh (ch cos [3 n n nn n nD ++ () (. . 3.1.3).. 3. 64- . -. , , [84].(3.33),n., . 3.1..(3.23)(3.34)(3.7),(3.2),.. , =+ =1) sin( ) ( ) , (nn Dntn nt e A x t x yn n (3.36)( Cn 1). =+ + =1)] sin( ) cos( [ ) ( ) , (nn n n n n Dn Dntn nt t e A x t x yn n &. (3.37)Ann(3.37),..,-.. (3.36) (3.37) t = 0 n = 0. [. (3.38)] , (3.6),==10sin ) ( ) (nn n nA x x y ;==10cos ) ( ) (nn n n nA x x y &. (3.38) m(x) 0 l. (3.25) , m = n, . . cos || || ) ( ) ( ; sin || || ) ( ) (200200 n n n nln n n nlnA dx x y x A dx x y x = = & (3.39)=ln ndx x y x I00 1) ( ) ( ;=ln ndx x y x I00 2) ( ) (& .(3.40) ( ). || || cos ; || || sin2221 n n n n n n n n nI A I A = = (3.41)( )22212|| ||1||.|

\|+ =nnnnnII A ; (3.42)3.1. 65nn nnII21arctg = . (3.43)(3.42)(3.43) (3.36) - . , .-(. 2.2, )..-(3.36),. , . =+ = =1) sin( ) ( ) , ( ) , (nn Dntn nt e A x EI t x y EI t x Mn n , (3.44) =+ = =1) sin( ) ( ) , ( ) , (nn Dntn nt e A x EI t x y EI t x Qn n . (3.45)) (xn ) (xn .3.1.(3.36) .,(3.36),,,(3.44)- , (3.45) .,, ) (0x y& x[53]. -S(). , , S /, , .. (-) .,-(3.1)(3.2)-.,- .. , . 3.2.4. -,., , - 0 , . , -, - .. 3. 66.,-[Cn (3.23)]. ) ( ) (~x C xn n n = . (3.46)(3.25),(3.40),(3.42)(3.43),, n- Cn ,n n nC A A =~, (3.47).(3.46)(3.47)(3.36), = + = + = == 1 1) sin(1) ( ) sin( ) (~) , (nn n nnn nnn n n nt ACx C t A x t x y =+ =1) sin( ) (nn n n nt A x . (3.48)(3.36),,.3.1.2. yxF(x,t) = Fmaxf(x)(t) xl. 3.2. -.,---F(x,t),..-, . 3.2. ) , (2244t x FtyxyEI =+ , (3.49) y(x,t) , -.,.- , :) ( ) ( ) , (maxt x f F t x F = ,(3.50) (t) f (x) -, 1, Fmax , /.3.1. 67- ) ( ) (max2244t x f FtyxyEI =+. (3.51).- :==1) ( ) ( ) , (nn nt x t x y , (3.52) n(x) n- , ; n(t) , -. (3.52) (3.51):) ( ) ( ) ( ) ( ) ( ) (max1 1t x f F t x t x EInn nnnIVn = + ==& &. (3.53) (3.9), - ) ( ) (2xEIxnn IVn = . (3.54) (3.53), ) ( ) ( )] ( ) ( [ ) (maxt x f F t t xnn n n n = +=12& &. (3.55) f (x) :) ( ) (1x a x fnn n == (3.56)(an).(3.55) ) ( ) ( )] ( ) ( [ ) (maxx a t F t t xnn nnn n n n === +1 12 & &. (3.57)n(x),, n(t) ) ( ) ( ) (maxt Fat tnn n n = +2& &. (3.58) :nna =. (3.59) ) ( ) ( ) (max2t F t tn n n n = +& &. (3.60)nn-,, .. 3. 68 (3.60) . -,. ) ( ) ( ) ( 2 ) (max2t F t t tn n n n n n n = + +& & &. (3.61) ) ( ) (maxt F tn n n = . (3.62)) ( ) ( 2 ) (2t t tn n n n n n = + +& & &. (3.63)- n - (t),(2.13). . (3.62)(3.52),- = =1max ) ( ) ( ) , (nn n nt x F t x y . (3.64).-an(3.56),-.(3.56) n(x) : ==10 0) ( ) ( ) ( ) (nli n nlndx x x a dx x x f . (3.65) (3.25), i n 0, i = n20|| || ) ( ) (n nlna dx x x f =.(3.66) =lnnndx x x f a02) ( ) (|| ||1.(3.67), n- = lnnndx x x f02) ( ) (|| ||1 . (3.68) , a x b(. 3.3,),- (3.68) 0, .. 0 l, a b.3.1. 69)xy F(x,t) = maxF f(x)(t) abl)P2(t) Pk(t) P1(t) a1xyakl. 3.3. : ; x = a1,, ak P1 (t),, Pk (t), , (. 3.3,), n - . [30], :1) = =; 0 ; 0 0) (xxx (3.69)2)1 ) ( = dx x . (3.70) (3.69) (3.70) , ) ( ) ( ) ( a dx a x x = . (3.71) - P1, , Pk x:) ( ) ( ) ( ) (2 2 1 1 k ka x P a x P a x P x P + + + = K .(3.72) (3.71) , (3.68) :+ + = + + lk n k n n k ka P a P dx x a x P a x P01 1 1 1) ( ) ( ) ( )] ( ) ( [ K K . (3.73),21 1|| ||) ( ) (nk n k nna P a P + += K. (3.74),,, (3.68) (3.74).. 3. 70.,n(x)-()n., . ) ( ) (~x C xn n n = . (3.75) (3.25) , 2 2 2|| ) ( || || ) (~|| x C xn n n = . , - nnnC = 1~. (3.76) (3.75) (3.76) (3.64) = ==1nn n nt x F t x y ) (~) (~) , (max == = =1max1max) ( ) ( ) (1) (nn n nnn nnn nt x F tCx C F . (3.77), ..- - (3.64). = = =1max ) ( ) ( ) , ( ) , (nn n nt x EI F t x y EI t x M . (3.78) = = =1max ) ( ) ( ) , ( ) , (nn n nt x EI F t x y EI t x Q . (3.79)) (xn ) (xn - . 3.13.1.3. . , - EI -) ( ) , (0t X t x F& & =y) (0t X& &x) (0t X& &xl. 3.4. ) (0t X& & (-). - -) (0t X& &(. 3.4). - - 3.1. 71) (02244t XtyxyEI& & =+. (3.80).--. :==1) ( ) ( ) , (nn nt x t x y , (3.81) n(x) n- , ; n(t) , -. (3.80):) ( ) ( ) ( ) ( ) (01 1t X t x t x EInn nnnIVn& && & = + ==. (3.82) (3.54) - , ) ( )] ( ) ( )[ (012t X t t xnn n n n& && & = +=, (3.83), ,) ( )] ( ) ( )[ (012t X t t xnn n n n& && & = += . (3.84)) ( ) (0t X x f& &, f (x) = 1 0 l f (x) = 0 . - f (x) :) ( 1 ) (1x x fii i = = = . (3.85) i, (3.85) n(x) : = =10 0) ( ) ( ) (iln i ilndx x x dx x . (3.86) (3.25) , i = n, 0, 2020|| || ) ( ) (n nln nlndx x dx x = = , (3.87) ||n||2 n- [. (3.26)]. = lnnndx x02) (|| ||1. (3.88). 3. 72(3.84),(3.85)(3.88)- ) ( ) ( )] ( ) ( )[ (012t X x t t xn nnn n n n& && & = +=. (3.89) n(x) . , n(t) ) ( ) ( ) (02t X t tn n n n& && & = + . (3.90)(2.18),,.-, n, -n-.,- . -, , ( ). , . ) ( ) ( ) ( 2 ) (02t X t q t q t qn n n n n n n& && & & = + + . (3.91) ) ( ) ( t t qn n n = , (3.92) :) ( ) ( 2 ) (02t X t tn n n n n n& && & & = + + . (3.93) (2.18).(3.92)(3.81),- = =1) ( ) ( ) , (nn n nt x t x y . (3.94)3.1.4. -,. (3.36) (3.64). :(3.44)(3.77), (3.45) (3.78).,,3.1. 73,-- (3.64). - (), .. .-(3.64).- , - , (3.36).3.1.5. (3.1)(3.2) ,-- . . , , . 3.1.,- , . , - , .,-- . -.. ( ). -[84].- 0 144 22 242244=+ ||.|

\|+ +tyGIt xyG fEItyAxyEI , (3.95) ; G ,) 1 ( 1 +=EG , (3.96) ; f , (. . 3.3.1).(3.95)(3.1),,,.([54, 84]).,. 3. 74,,... [84]. , (3.44)(3.45), , .(3.95)- ., , ,-.,,- -., , 1/8 1/10 .,,-,.l,b = 1.- I. lIi=+. (3.97),-p-+(..9.1). ( ) , , 2 ++= pi . (3.98)- [53]:==1sin sin1 4) , (nntlnxnEJlIt x M , (3.99), (3.97) (3.98),=+=1sin sin12) , (nntlnxnEJi pt x M . (3.100)-100T1/32,T11.- 3.1. 7513213T t = . (3.101) : EJi pM+=max2477 . 1 . (3.102)(3.102). EJl21|.|

\|= . (3.103) 816477 . 1213maxl pM=+ . (3.104) p:82l pM= . (3.105) , 1 13max762 . 016477 . 1 M M M + += = . (3.106) :1max762 . 0 += =MMk . (3.107),(3.101). (3.107). . 3.5 ,.. 1 +k. (3.108). 3.5- , . -- x l, ( -[53]).100.,. 3. 76. , , , t = 10T1/32. 0 0.5 x/l0.20.20k/(1+)T1/320 x/l0.500.20.22T1/32 k/(1+)00.2k/(1+)00.5x/l4T1/320.40.20k/(1+)6T1/320 0.5 x/l0.40.20k/(1+)8T1/320 0.5 x/l0.40.20k/(1+)10T1/320 0.5 x/l0 0.5 x/lk/(1+)0.60.40.2012T1/320 0.5 x/lk/(1+)0.80.60.40.2013T1/3200 0.5 x/lk/(1+)0.60.40.214T1/320.40.200 0.5x/lk /(1+)16T1/32. 3.5. : - ; ;T1 1 3.2. 77,--..3.5,,,-, , - .3.2. 3.2.1. , , - - . , -,-,-(, ). - , -.- -,-, .3.2.2. -.--h,- xy(. 3.6),- 022=+ tww D , (3.109)w(x, y, t)-, hw(x,y,t) zxxayb. 3.6. x y t; ,2222y x += ; (3.109) ,/2;D,) 1 ( 1223 =EhD ; (3.110). 3. 78 E ; .., .-,(),-,.-,,[22, 63].(-),(- ), ( -)., .,- t = 0:) , ( ) 0 , , (0y x w y x w = ; ) , () 0 , , (0y x wty x w&=. (3.111).(3.109)(). , , , , . .()- ==+ =1 1) sin( ) , ( ) , , (m nmn mn mn mnt A y x t y x w , (3.112) mn(x,y); mn;Amn; mn ..,,.,, , . by nax my xmn sin sin ) , ( = , (3.113) Dbnammn||.|

\|+ =22222.(3.114)- , , [63].3.2. 79 [22].1. :) :, 0 ) , ( ) , (0 0= a bij mndxdy y x y x (3.115)m, n- i, j;) :

||.|

\|+ a bijmnijmnij mnx y y xD0 02222222221) 1 ( 2 022=)`(( dxdyy x y xijmn. (3.116)2. ( ).,f (x,y)0 x a, 0 y b mn(x,y).3.2.3. .-R(x,y,t)(-) ) , , (22t y x Rtww D =+ . (3.117) , ) , ( ) ( ) , , (2 1 maxy x f t f R t y x R = ,/2, (3.118)Rmax,;f2(x,y),1/2,-, 1; f1(t) ,,*.,S1, (. 3.7). (3.117) ) , ( ) (maxy x f t f Rtww D2 122=+ .(3.119)

* ,. 9.7,,- 110 , , Rmax = 110 , f1(t) - .. 3. 80 ,1 ) , (12=Sdxdy y x f .(3.120),, f2(x,y) =. ,; ,) , (11 12 0 1S x,yS x,y Sy x f(1/2) (3.121)w(x,y,t)zxR(x,y,t)yxabS1f2(x,y)S. 3.7. -.-(3.117),,- ,,.f2(x,y),--.,- .,.- n . . 2.2.(3.119)- (. . 4.1.3).3.2.4. zxw(x,y,t) ) (0t Z& &) (0t Z& &) ( ) , , (0t Z t y x R& & =. 3.8. --(. 3.8)(-).) ( ) , , (0t Z t y x R& & = , (3.122)) (0t Z& & . (3.122) (3.117): ) (022t Ztww D& & =+ . (3.123)3.2. 81 . ===1 1) ( ) , ( ) , , (m nmn mnt y x t y x w . (3.124) mn(t) ) (02t Zmn mn mn& && & = + , (3.125) mn ,= ) () , (|| ||1Smnmnmndxdy y x ,(3.126)=) (2) , ( || ||Smn mndxdy y x . (3.127)(3.126)(3.127) S. ) ( ) ( t tmn mn mn = (3.128) == =1 1) ( ) , ( ) , , (m nmn mn mnt y x t y x w , (3.129) mn(t) - mn,- ) (0t Z& &. mn(x, y) [. (3.113)]. -,,- x y:y n x m mn , , = . (3.130) , - .- -. .-, , .. 3. 823.2.5. h--,- -.(.. 3.1.5), . , - [22].3.3. -,,..,.- .3.3.1. . 3.1.,-(2.3)(2.6),,. 4.1.. 4.2,-,mk . mk.3.2, : E G ;I;F-; m ; f , :1.3;1.11;2;, ..- F , , F.3. 2. k, / m, lmll fGF0.405m3.3. 83,,,[111],- .3.3.2. - mEIlKfb212= , , (3.131) l , ; E ,/2;Ib , 4;m,/;K.2-3-. 3.9-:2--K. 3.11;3-, l(),l l1 l2,K. 3.10.l l1,K2-,- l1 ;4-,,K, -l l1l l12- ll1l2l l1 l2 3- . 3.9. 2- 3- 0 0.2 0.4 0.6 0.8 l1/l33.544.5K3- l1/l2 = 21.51 2-. 3.10. K 2- ;4-,,- , K , 3- , - .. 3. 84 m EI - 80 % , (3.131) - .3.3.3. yxb h1 = hh3

bh3h1h2 bh3h1h2h. 3.11. (. 3.11) - MMhbf211= , ,(3.132) b , ; h ,32 1 h h h + = ;(3.133)M , h3,31 3h h = ;(3.134)M;1,: 1 = 700; 1 = 1000.3.3.4. 50 h f 461 = , , (3.135) h , [111]. . , , -, ().,[111],,,- .3.3. 853.3.5. ,,, 1(x)- ( ) h x x = ) (1.(3.136)- . 3.12.- . 3.3.0.20.40.60.8x/h0 0.2 0.4 0.6 0.8 1(x) = 1.510.6 = 2.5 2. 3.12. 3. 3. 2.5 2.0 1.0,- 0.6,,- 1.53.3.6. (, -,;.. 12.3)m(x)m. 1 =lldx xdx x x mm021021) () ( ) (,(3.137) 1(x);l.,m, m = m.-m h3[. . 3.9- (3.132)].l,- m(x), m. 3. 86m(x)l / 3- , 1(x).3.4. -,.- , - . .,[6] , -,.-,,-,- . [6]-:0.6;- () 0.3. -0.01 0.10.11n F/ = 10.750.500.250. 3.13. F:- 0.6; () 0.3, 0.2.- ,.- , - . [127, 135], I ( ) I:( )||.|

\|+ = + = FbdbhI I I12 21213 , (3.138) b h ; d -;F , . 3.13. 3.4. 87 n, n Ea E ,EEn = ; (3.139),..-- . F 2 23) 1 . 0 ( 9 . 1 ) 1 (3+ + = K K nKF ,(3.140) ;q m m K 22+ + = ; (3.141)((

+ = 9 . 1 1 n m ;(3.142)((

+ = 19 . 0 1 n q . (3.143) , - [61]. - . 4 -,-. - . , .4.1. 4.1.1. - - , . 2, .,..-.(- ). , , ,. 2, . -,,-,, .-,. - :) ( t F y k y m = +& &,(4.1)y;m,kF(t),.-, [. (2.2)]. , , - .4.1. 89- , , .- . . 5.4.1.2. (-)[37],-.4.1.-l- EJ ; , -:mk.:F(t) p(x)(t). -[37],, , . - - , x t: ) ( ) ( ) , (x f t y t x y = .(4.2)y.-- QyTyTdtd= &, (4.3) T ;Q.xp(x)(t)y(x,t) = y(t)f(x)ymkablF(t)y. 4.1. - 1 | | | | = + =202) , (21) , (21t l y m dx t x y Tl& &| | | |22022) ( ) (21) ( ) (21t y l mf dx x f t yl& &+ = . (4.4).4. 90(4.3)., (4.3) . -, ((

+ =) ( ) ( ) (202l mf dx x f t yyTdtdl& &&. (4.5)Q.-y.- y(x, t) ) (x f y y = . (4.6) F(t) p(x)(t) = + =ldx t x y t x p t b y t F A0) , ( ) ( ) ( ) , ( ) ( ((

+ =ldx x f x p t b f t F y0) ( ) ( ) ( ) ( ) ( . (4.7) [y(a, t)+y(a)] y(a, t)( ) | | ( ) | |222) ( ) , ( ) ( 22) , ( ) ( ) , (2a y t a y f ykt a y a y t a ykAk + = + = . (4.8)- , . , ) ( ) ( ) ( ) ( ) (2 a f t ky y a f t y a f y k Ak = = . (4.9) y(x,t)| | =ldx t x y EJ A02) , (21. (4.10)x., ,( ) ( ) | | + =ldx t x y y t x y EJ A02 2) , ( ) , (21 | | = = l ldx x f EJ t y y dx t x y y EJ02 0) ( ) ( ) , ( 221 . (4.11) , | |)` + = l ldx x f EJ t y dx x f x p t b f t F y A0 02 ) ( ) ( ) ( ) ( ) ( ) ( ) ( . (4.12) :4.1. 91((

+ + = l ldx x f EJ a kf t y dx x f x p t b f t F Q02 20)] ( [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( . (4.13) (4.5) (4.13) (4.3), =((

+ +((

+ l ldx x f EJ a kf t y l mf dx x f t y02 2202)] ( [ ) ( ) ( ) ( ) ( ) ( & &+ =ldx x f x p t b f t F0) ( ) ( ) ( ) ( ) ( .(4.14),-) ( ) (202l mf dx x f ml+ = , (4.15) + =ldx x f EJ a kf k02 2)] ( [ ) ( ,(4.16) + =ldx x f x p t b f t F F0) ( ) ( ) ( ) ( ) ( . (4.17) EJx Mx f) () ( = . (4.18) :+ =ldx x MEJa kf k02 2)] ( [1) ( .(4.19) 1.--l-,EJ,--(. 4.2).l = M.f(x) ylxp(t). 4.2. .4. 92|.|

\| =lxx f 2cos 121) ( .,.. . (4.15) :M l dxlxdx x f ml l8383 2cos 141) (0202= =|.|

\| = = . (4.16):34024022 2cos241)] ( [l EJdxlxlEJ dx x f EJ kl l =|.|

\|= = ., (4.17):) (22cos 1 ) (2) ( ) (0 0tpldxlxtpdx x f t p Fl l =|.|

\| = = .,. 4.1,- , - , . :1) s ks, - xs; r mr xr; n Fn(t) xn;2) - xd, I0d;3) - x;4) (), k(x) (, );5) , , Mj(t), - xj. [37]:| | + + =20202) ( ) ( ) ( ) (d d r rlx f I x f m dx x f x m , (4.20) + + =l ls sdx x f x k dx x f x EJ x f k k0202 2) ( ) ( )] ( )[ ( ) ( , (4.21)4.1. 93 + + =l ls sdx x f x k dxx EJ x Mx f k k02022) ( ) () ()] ( [) ( , (4.22) + + = ) ( ) ( ) ( ) ( ) ( ) ( ) (0 j jln nx f t M dx x f x p t x f t F F . (4.23).,(4.1)[.(2.2)].xm- m, , -(x)(,), [37]+ =lm mdx x f x x f02 2) ( ) ( ) ( . (4.24)---,,--h(x,y)(.4.3).xi,yimi. -ax byw(x,y,t)w(t). 4.3. - p(x,y)(t),Fn(t)xn,yn.w(t), ) , ( ) ( ) , , (y x f t w t y x w = .(4.25)f (x,y),,-,- .- , [37]. + = ) , ( ) , ( ) , (2 2 i i iSy x f m dxdy y x f y x m ;(4.26) .4. 94 (((

||.|

\| ||.|

\|+=Sdxdyy x fyfxfyfxfD k2222222222) 1 ( 2 ,(4.27) D ,) 1 ( 1223 =EhD , (4.28) E , . + =Sn n ndxdy y x f y x p t y x f t F F ) , ( ) , ( ) ( ) , ( ) ( . (4.29) (4.26)(4.29) S., .,-, , EJlEJlmk22279 . 2234= = = . 2 % ( - 22.37). -, .,(4.1)-,,-y,(4.2)-,.,f (x).y, , - f(x), . .f(x),.[.(4.18)],1.,,f(x),,. , -.,.4.1. 95 , y, (4.1),.-,y (x,t),(4.2), . , f (x).,,,(4.18)-,(4.19),- , .,,-,.-,-,, -. , - f(x) , 1.,,, , . f(x), , .4.1.3. -- .- , . 3.2.3 (. 3.7). -(3.118). (3.118) (3.119), -.S. S1 ) , ( ) ( ) , , (2 1 maxy x f t f R t y x R = ,/2, (4.30)Rmax,;f1(t)-,-; f2(x,y) , 1/2, ,1.,- S1 f2(x,y) - , =. ,; ,) , (1112 01S x,yS x,ySy x f (1/2) (4.31).4. 96., w(x, y, t) :) ( ) , ( ) , , ( maxt T y x w R t y x w = ,(4.32)w(x, y)-; ) , ( ) , (2 y x f y x w D = ;(4.33)T(t) , - .(4.32),(3.124)-.(-) :( ) ( ) ( ) ( ) ( ) ( ) ( ) | | y x f t f t T x,y w t T x,y w D R t y x L , , ,2 1 max + =& & ,(4.34), (4.33),( ) ( ) ( ) ( ) ( ) ( ) ( ) | | y x f t f y x f t T t T x,y w R t y x L , , , ,2 1 2 max + =& & .(4.35),,L(x,y,z)- , .. =) (0 ) , ( ) , , (Sdxdy y x w t y x L , (4.36)S. (4.35):( ) ( ) ( ) ( ) = + ) ( 2) (2) , ( ,S Sdxdy y x w y x f t T dxdy x,y w t T & &( ) ( )=) ( 2 1) , ( ,Sdxdy y x w y x f t f .(4.37),- T(t):( ) t f T T122 = +& &,(4.38) ,( ) ( )()( )()=SSdxdy x,y wdxdy y x f x,y w22 2, . (4.39)(4.38).,-4.1. 97 ,,..(,-).- - .f2(x,y)(4.31),- ( )()=SSdxdy x,y wdxdy x,y wS) () (21211 . (4.40) , S1 S,, (x0,y0):) ( ) , (0 0 ,y x w y x w . (4.41) ()=Sdxdy x,y w,y x w) () (20 0 2 .(4.42) , - (x0, y0). , . - 0x, x1 x2 - . ()=Sxxdxdy x,y wdx x wx x) () (121 2221 . (4.43)(4.39),(4.42)(4.43) 1 . . , (4.38) :) ( 212 2 t f T T T = + +& & &,(4.44). , , , .-.4. 98,- ( = 0.050.10). . w -S1 - . -,(4.40) S1 1 :=) () , (111Sdxdy y x wSw . (4.45) , , (4.41), , ) , (0 0 y x w w . (4.46)k 1 , ..1wk = . (4.47) m :2km = . (4.48) F (4.29)( ) ( )=) (max) , ( ,Sdxdy y x w y x f R t f F 2 1 .(4.49)(4.39),, -,,.-,, . -,(,SCAD Office [35]) - .4.1.4. ---4.1. 99 ( , ..),-.-. .1. - -, .,- , . 4.1.2. -,. , ,-,- .--,. 4.1.3,- - . .,-,,- (,,).(,,..)-(.. 16.3).- .2. -,. ,.- . 3.1.5 -,.--, .4. 100 . , . 3.5,.,-(- 1 ) . --, . 3. -,-. -(,,), . ,- . -:,-,- . 4. 1. -x1x2 == x1+xxn-1 == x n-2+xxn == xn-1+ x xy1f(x1,y1) f(x2,y1) f(xn-1,y1) f(xn,y1) 1(y1)y2 == y1 + yf(x1,y2) f(x2,y2) f(xn-1,y2) f(xn,y2) 2(y2) yk-1 ==yk-2 + yf(x1, yk-1) f(x2, yk-1) f(xn-1, yk-1) f(xn, yk-1) k(yk-1)Yk == yk-1 + yf(x1,yk) F(x2, yk) f(xn-1, yk) f(xn, yk) k(yk), , =k nyyxxdxdy y x f I1 1) , ( . - x, y - f (x,y). . 4.1 nk ( ).- i(yi), - f (x,yi) x -y = yi.4.1. 101.,-, 1(y1) = [f (x1,y1) + 2f(x2,y1) + + 2f (xn-1,y1) + f (xn,y1)]x/2;2(y2) = [f (x1,y2) + 2f (x2,y2) + + 2f (xn-1,y2) + f (xn,y2)]x/2;.k(yk) = [f (x1,yk) + 2f (x2,yk) + + 2f (xn-1,yk) + f (xn,yk)]x/2.,y , .. :I = [1(y1) + 22(y2) + + 2k1( yk1) + k(y k)]y/2.,-, .. x, y.4.1.5. . 4.2m,kF- .. - - (. . 3.1.6). 4. 2. f (x), m, k, Fxyf(x)mF(t)0.5lly |.|

\| |.|

\| =l xllxlxlxlxlxx f2 1 4 1 320 4 3333) (l m 486 0. = ; 348lEJk = ; F = F(t)lyf(x)F(t)0.5lxym |.|

\| |.|

\| =l zllzlzlzlzlzx f2 1 16 1 1220 16 12) (3 23322l m 371 . 0 = ; 3192lEJk = ; F =F(t).4. 102 . 4.2 f(x), m, k, Fyf(x)mlxyF(t)0.5l |.|

\| + =l zllxlxlxlxlxlxx f2 1 2 2 5 320 5 333333) (m =0.445l; 371 109lEJk . = ;F =F(t)yf(x) mF(t)lxy||.|

\|+ =333 221lxlxx f ) (m =0.236l; 33lEJk = ;F =F(t)yf(x) mp(t)0.5llyFx||.|

\|+ =44332516lxlxlxx f ) (m = 0.504l; 315 49lEJk . =;F = 0.64pl(t)lyf(x)0.5lymp(t)Fx2221 16 ) (|.|

\| =lxlxx fm = 0.406l; 38 204lEJk . =;F = 0.533pl(t)yf(x) m0.5llyp(t) Fx||.|

\|+ =44332 3 4lxlxlxx f ) (m = 0.483l; 32 115lEJk . =;F = 0.6pl(t)f(x)mlyp(t)Fxy||.|

\|+ =444 331) (lxlxx fm =0.257l; 32 3lEJk . =; F = 0.4pl(t)4.1. 103 . 4.2 f(x), m, k, Fbyaxy x f sin sin ) , ( =m = 0.25ab; 22 241 14|.|

\|+ =b aDabk F ( ): F(t), , F = F(t);aba/2b/2yxF Db af|.|

\|+ =2 21 12, p(t), - ,F = 0.405abp(t);|.|

\| |.|

\| =byaxy x f 2cos 12cos 141) , (ab m 649 =||.|

\|++= 606 0723 02 24 4 4..b ab aabDk F (- ): F(t), - , F = F(t);baa/2b/2yxF , ,||.|

\|++= 606 0134 12 24 4..b ab a Dabf p(t), , F = 0.25abp(t):l; , /; EJ ; F - , ; q , /. ,-f(x),. ,, 1. y(x) = yf(x), -.,y-.. 4. 104:D-, [.(4.28)]; ,/2.w . f(x,y) . - m F (4.26) (4.29), k - f ,[63, 84].F. w. , w.4.2. 4.2.1. , .(4.1),-- Rmax, T(t). -, k,T(t).- ..4.4-. . 4.3 [78]. ,.4.4.- , . , , [26,61,68, 78,135].- . 8 .4.2.2. , . 4.2.1,-,. 4.4.-.4.4;f (t)( ), (t) ; . 0 t (..-) (t) [78], 4.2. 105 . t (.. -)-,.- ((

++ =) ( sin) ( ) () ( cos ) ( ) ( t t e tDDDt&, (4.50) ; D .0 t1t 03t 0200.511.5k0.5 1 1.5 2 2.5 3 t 05t 060.5 1 1.5 2 2.5 3 0.51t 0401.5k0.5 1 1.5 2 2.5 3 00.511.5k1/ = 0.5 0.4 0.3 0.2 0.1 0tp/20p1. 4.4. T = = ) (~2 , ; T . 4. 1064. 3. ~ k15 . 0~0 ~sin 2~5 . 0 22.795 . 0~0 )~2 ( )~2 cos 1 ( )~2 sin~2 (2 2 + ;~795 . 0 )~2 ( 1 1 + = k3375 . 0~0 )~2 ( )~2 sin~2 ( )~2 cos 1 (2 2 + ;~375 . 0 | | )~2 ( )~2 (arctg 1 2 45 . 0~0 )~4 1 ( )~cos~4 ( ;5 . 1~5 . 0 |.|

\|++ 1~22sin~21~2~4sin~4 1~2 ;~5 . 1 |.|

\| 1~22sin~21~2~4sin~4 1~2 55 . 0~0 ( ) )~(~cos 1 2 ;2~5 . 0 )~( ]sin )~sin( 2~2 [ + ;~2 )~( 2 1 + , cos 1 )~cos( 2 + = ; )~sin(~sin = ;1 = ~703 1. = 65 . 0~0 )~1 ( )~(sin ;2~5 . 0 { } )~1 ( )]~cos( 1 [~cos 12 2 ;~2 ) 1~(~2 2 4. 4. (t) - f(t) 0 t t - (4.50), 1 1 t cos 1 cos =1 ) ( ; sin = ) ( & .2t t t sin ) ( ) (sin 1 ) ( = ; ) cos ( = 1 ) (& .4.2. 107 . 4.4 (t) - f(t) 0 t t - (4.50), 3 t 1 tt t cossin + 1 cos ) (sin ) ( = ; sincos 1) ( + + =& .4tsin|.|

\| 2sin sin) (2 2tt ( ) | |( ); ) 2 sin( sin) ( ) (2 2 =( ) | |( ). ) 2 cos( ) 2 ( cos) ( ) (2 2 2 =& 5 6 . 4.4 (t) - : 5: =ttttt f222;20 2) ((4.51)( )|.|

\| =|.|

\| =((

|.|

\| + =. cos cos ) (, sin cos ) (; sin sin ) (; sin) (21242 214 (4.50), 2222220 &tt t t tt t tt(4.52) 6: |.|

\| =tt f2cos 121) (; (4.53)( )==|.|

\| = sin) ( 42) (, cos 1) ( 42) ( (4.50), ] ) ( 4 [ 22cos 1 ) ( ) cos 1 ( 40 ) (2 222 222 22 2&ttttt(4.54). 4. 1084.2.3. - ( ) ,. 4.1.-., . .2. Lear Jet. - . 8. -, = 54 1/(8.6 ), = 0.07. -Rmax = 12 (. . 8.14,);-(. 8.14,)k = 1.25.,- 15 25 1 12 = = = .maxR k F.4.2.4. n - . 4.1.4.,,. . 2, . 3. . . 2, j- ,.. j(j--).R(t)Rmax,- = == =njj jnjjt q t x t x1 1) ( } { )} ( { )} ( { , (4.55){j(t)}j-;{j}j-;qj(t)(),- (2.75). k( i),,4.2. 109qi,max,R(t)k( i)Rmax.(2.75), . max 2 2max ,) (|| ||} { } {R kJqii iTii = . (4.56) (4.55) i- :max 2 2max ,) (|| ||} { } {} { } { R kJxii iTii i = . (4.57), -R-, R J x K } { } ]{ [ = . (4.58) : = == =njj injja x x1 1} { } { } { . (4.59)ai,(4.58) i- :R J a KTinjj iTi} { } { } { ] [ } {1 ==. (4.60) (2.37) , RJai iTii2 2|| ||} { } { = . (4.61) (4.61) (4.59), , (4.57).(4.58)k( i)q i,max-R(t).-.- max , iq& & .- , ) (ik .,-,. 4.3.[12], ) ( ) (2 i i ik k = . (4.62) (4.57), .4. 110max2max ,) (|| ||} { } {R kJqiiTii=& &. (4.63) max2max ,) (|| ||} { } {} { } { R kJxiiTii i =& &. (4.64)(4.64)[M],,- i- ,max2max , ,) (|| ||} { } {} ]{ [ } ]{ [ } { R kJM x M FiiTii i i = =& &. (4.65)-{xi}{Fi,}(,,-, ).-.,,,.,--(,),., . 4.3.4.3. 4.3.1. (. . 2.4). , , . (,-, ), , - (, , ). --()(Re-sponseSpectrumMethod). [12], -. - .4.3. 111() (),(-) (). -,, .,-,--,-.--()(-re-sponsespectra).- , ,-,- .- [12], -(-, ) . 17.)Amaxf 0Sa ( f, )12 > 1f)f0Sd ( f, ). 4.5. : ; Sa(f,). . 4.5,. f = 0 . , - . , -- . .,-f,-,- Amax. , , .., . f, - ,.. ,- , , . - , -,AmaxZeroPeriodAc-celeration (ZPA), .. . - AZPA, fZPA..4. 112 -,,,f = 0, f f Amax. Sd( f, ). . 4.5,: f = 0 (- ), .Sa ( f, ),Sv( f, )Sd( f, ) [12]:) , ( ) 2 ( ) , ( 2 ) , (2 f S f f fS f Sd v a= = , (4.66) f , . Sa( f, ) , , ,-f-,-Amax.-,Amax.Sd( f, )Sv( f, )- .Sa( f, )--,- .4.3.2. - ,,-. (..2 3), . :1. .2. (..- ) (,,,- .), .3. - () .4. ,-,,,- .4.3. 113 , - , . . (1 ) . 2 3. .4.3.3. ,Sa(f, )-- f . /2, , m (),) , ( f mS Fa= , . (4.67)- . j- - -(.. 2), - . Fxmk. 4.6. , - -,, , .- . . 2.- (2.115). :==njj jt t x1) ( } { )} ( { , (4.68) {j} j- ,j j j = } { } { , (4.69){j}j-;j(t)j-fjj;jj-,, ,} ]{ [ } {} ]{ [ } {jTjTjjMJ M = , (4.70) [M] ; {J} , - ..4. 114 (4.68) j- . Sd(f, ) .- j- ) , ( } { } {max , j j d j jf S x = . (4.71)(4.68), ==njj jt t x1) ( } { )} ( { & & & &, (4.72) j- . , , j- :) , ( } { } {max , j j a j jf S x =& &, (4.73)1- F1,3F1,2F1,1F2,12- F2,3F2,2F3,23- F3,3F3,1xym1m2m3. 4.7. 3 { }== =njj jt M t x M F1) ( } ]{ [ )} ( ]{ [ & & & &, (4.74)j-- (.. j- ):) , ( } ]{ [ } {, j j a j jf S M F = .(4.75) . (2.26) :} ]{ [) 2 (1} ]{ [2jijKfM = . (4.76) (4.66) ) , ( } ]{ [ } {, j j d j jf S K F = .(4.77), (4.71)..4.7- . 4.3. 115 Sa(fj,j), () . -().-(-).j- {Rj} (, , - ). Rjk , - j- k- .,- (, .) ,...- .-.. (3.94) := =1) ( ) ( ) , (nn n nt x t x y , (4.78)n(x)n-;n(t)n-- fn n;n - n- ,= lnnndx x02) (|| ||1; (4.79) ||n||2 ,=ln ndx x02) ( || || . (4.80)- . 3.1. n- n n nx x = ) ( ) ( . (4.81) ==1) ( ) ( ) , (nn nt x t x y . (4.82)(4.82)-.n- :) , ( ) ( ) (n n d n nf S x x y = . (4.83).4. 116 - . ) , ( ) ( ) ( ) (n n d n n nf S x EI x y EI x M = = , (4.84) ) , ( ) ( ) ( ) (n n d n n nf S x EI x y EI x Q = = , (4.85)n n nx x = ) ( ) ( ;n n nx x = ) ( ) ( . (4.86)) (xn ) (xn - . 3.1.- . , . 3.2.4 - == =1 1) ( ) , ( ) , , (m nmn mn mnt y x t y x w . (4.87), ) , ( ) , ( ) , (mn mn d mn mn mnf S y x y x w = . (4.88) - .4.3.4. () --,. (-),.- , ,-. . k- Rk, j- Rjk. , ,j-,,..-j-.Rk = Rjk. . . : = =sjjk kR R1| | , (4.89)4.3. 117 s , n(). , --., .(4.89) Rk. [12] - . ():= =sjjk kR R12. (4.90)Square-Root-of-the-Sum-of-the-SquaresMethod(SRSS)., 10 % (, - ). , 10 %(), -.10 %, Ten Percent Method. - = = =+ =mlmrrk lksjjk kR R R R1 1 12| | 2 , (4.91) -, , -, f. -CQC(-CompleteQuadraticCombination).- , , = = =sjskk j jk kR R R1 1 , (4.92). 1 ;) 1 ( 4 ) 1 () 1 ( 82 2 2 22 3 2 =+ + +=kjjkffrr r rr r jk = kj . NRL [128] (- NRL US Naval Research Laboratory):.4. 118=+ =sm jjjk km kR R R12,(4.93)Rkmk-s-, m- ; Rjk sm .- .-,.,-. , - .-, - ( , ). , - , . ,,-, . , - , - .4.3.5. s,-, n, ..(,- .), . --,: , , -.,-, . 2 .1. - . , - , .,(4.83)- s . -4.3. 119,-, . 4.3.4. , -,:--.--.- . , , -. 3.1,20 ,.,-,-.s,-(- )==sjj d jf S x x y12max)] ( ) ( [ ) ( . (4.94)(.. , ), , .(3.85), (4.81) 1 ) (1==xjj . (4.95), s ( ): = == = =sjjsjjjjx x x x1 1 1) ( 1 ) ( ) ( ) (~ . (4.96)f fs,.. (4.94), frSd ( fr ),-, ,) ( ) (~) (~max r df S x x y .(4.97) -,- ( +)212max)] ( ) (~[ )] ( ) ( [ ) (r dsnj d jf S x f S x x y + = =+. (4.98).4. 120, ) ( ) ( ) (max max maxx y x y x y+ . (4.99)(4.66) Sa(fj ):==sjj j a jf f S x x y12 22max] ) ( ) ( [) 2 (1) ( , (4.100) | | | |2 212 22max) ( ) (~) ( ) () 2 (1) (r r asjj j a jf f S x f f S x x y + ==+. (4.101)) (maxx y+,, f > fs 2max ,) 2 ( ) ( f S f Sa d , (4.102) Sa,max -.fr = fs+1,..- , 21 max , max) 2 ( ) (~) (~+s af S x x y , (4.103)

2 21 max ,12 22max] ) (~[ ] ) ( ) ( [) 2 (1) (+=++ = s asjj j a jf S x f f S x x y . (4.104)) (maxx y+ , -Sa,max .- (, ..). , . .- - [. (2.82)] = == =njj jnjjt t x t x1 1) ( } { )} ( { )} ( { , (4.105){ j}j-,(2.81); j( t ) j- ; n .j- Sd ( f i ):4.3. 121) ( } { } {j d j jf S x = . (4.106) , . ----. , .s(n)., ( , ). k- ==sjj d jk kf S x12)] ( [ . (4.107) {j} (2.114), :} { } {1Jnjj == ; (4.108){J}(2.107).-}~{ { j}, (s +1)-: = + = = =sjjns jjJ1 1} { } { } { }~{ .(4.109)f fs,..(4.107),frSd(fr),-, - ,) ( }~{ }~{r df S x , (4.110) k- ) (~ ~r d k kf S x , (4.111) k~ k- }~{ . , ( +) = =++ = + =sjr d k i d jksjk j d jk kf S f S x f S x12 212 2)] (~[ )] ( [~)] ( [ . (4.112) (4.66) kx +kx - Sd(fj), Sa(fj):| |==sjj j a jk kf f S x1222) () 2 (1.(4.113).4. 122| | | |221222) (~) () 2 (1r r d ksjj j a jk kf f S f f S x + ==+.(4.114) ,- (4.103), 212 ) (~ ~max , +s a k kf S x , (4.115)Sa,max f fs. | | | |221 max ,11222~) () 2 (1+=++ = s a ksjj j a jk kf S f f S x .(4.116) Sa,max , +kx .2. -. .(4.84).- :) ( ) ( ) ( ) ( ) (j d j j j d j jf S x EI f S x EI x M = = . (4.117) ) (xj j . 3.1, (4.108), (4.106) Sd( fj ) Sa( f j ), lx jf S ljx Mj ajjsin ) (] 1 ) 1 [( 2 1) (23 3 = . (4.118), . ==sjjx M x M12) ( ) ( . (4.119).,,f fs fr. = = ) ( ) (~) (~j df S x EI x M = = = ||.|

\| =sjj dr dj r d jsjjf Sf Sx M f S x EI1 1) () () ( ) ( ) ( . (4.120) ) (~) ( ) (212x M x M x Msjj =++ = . (4.121)4.3. 123.- ) ( ) ( ) ( ) (j d j j j jf S x x y EI x Q = = . (4.122) (4.118), lx jf lSjx Qj ajjcos ) (] ) 1 ( 1 [ 2 1) (2 2 = .(4.123) ==sjjx Q x Q12) ( ) ( , (4.124) ) (~) ( ) (212x Q x Q x Qsjj =++ = ,(4.125)==sjj dr djf Sf Sx Q x Q1) () () ( ) (~. (4.126),-n.- N -, , .j-{xj,max}-(4.71).-, , [T]:) , ( } ]{ [ } ]{ [ } {max , j j d j j jf S T x T N = = . (4.127)k--Njk.Nk, , . n , s , ==sjjk kN N12. (4.128)}~{ ,(4.109).- ) ( } { } { ] [ ) ( }~]{ [ }~{1r dsjj r df S J T f S T N||.|

\| = == , (4.129).4. 124 Sd ( f r ) f fs, fr.s.- (4.129) [T]:=||.|

\| ==) ( } { ] [ } ]{ [ }~{1r dsjjf S T J T N ) ( } {) (1} ]{ [1r dsjji df S Nf SJ T||.|

\| ==. (4.130) k- , =++ =sjk jk kN N N12 2~. (4.131), , , (4.130)21max ,) 2 () (+=sar dfSf S, (4.132)Sa,max,- ,3. .,,, . ..j- = lj jdx x MEI02) (21, (4.133) Mj(x) , (4.118). - , ) (] 1 ) 1 [(26265 2j ajjf Sj EIl = . (4.134) j 0. - , , -.- :4.3. 125== 1121 2465 2) 1 2 () (4jj ajf SEIl. (4.135) s , - == sjj ajf SEIl1121 2265 2) 1 2 () (4. (4.136) ( )+ == 1121 2465 2) 1 2 () (4 ~s jj ajf SEIl. (4.137),,f2j1 > fsSa(f2j1) - Sa,max. + == 16 62max ,5 2) 1 2 (14~s jaj EIS l. (4.138) [16]:001145 . 1! 6 2) 1 2 () 1 2 (1) 1 2 (136 61616= =< =+ =Bj jj s j, (4.139)B3 = 1/42., (4.138) , ..EIS la62max ,5 24~= .(4.140)-,,Sa(f2j1) , f > fs.~ , - 4max ,1121 2465 2) 1 2 () (4asjj aSjf SEIl+= =+. (4.141)n.- j- ( - )| | } ]{ [ } { ) (21} ]{ [ } {212jTj j d jTj jK f S x K x = = , (4.142).4. 126[K].(4.66)-(2.36)(2.37) :} ]{ [ } {) 2 () (21} ]{ [ ) 2 ( } {) 2 () (2122222jTjjj aj jTjjj ajMff SM fff S =(((

= . (4.143) j- * } ]{ [ } {jTj jM = . (4.144) (4.69) , } ]{ [ } {2J M mTj j j j = = . (4.145) (4.144) jjj ajff S22) 2 () (21= . (4.146) s < n . . , := = sjj12.(4.147) , n.- (2.36) (4.108), } ]{ [ } { } { ] [ } { } ]{ [ } {1J M M MTjnijTj jTj = ==.(4.148) , } ]{ [ } { } ]{ [ } { J M MnjTjnjjTjnjj||.|

\|= = = = = 1 1 1 .(4.149) (4.108):TnjTjJ} { } { ==1 . (4.150)

*()j-,..mj = { j}T[M]{ j} = || j ||2,- [. (2.73) (2.110)].4.3. 127c1M J M JTnjj= ==} ]{ [ } { ,(4.151) M .,n.,(s+1)-n-,-, = + = = =sjjns jjM M1c1 ~. (4.152)- MfSsa~) (~max ,2122 21+= . (4.153)~ , 212 + = =+~sjj. (4.154)-.-, , c~M M . (4.155) , .. .4. . :1) ;2) ;3) ;4) .,,- , , - . , - , , - . - .5. {j}. .4. 128{j},..,(2.79).- , .{j},(4.108).{xj}- Sd ( fj ) Sa(fj ). {j} :} {) () 2 (} {) (1} {2jj ajjj djxf Sfxf S = = . (4.156)6..(4.108)-.. [46](-).,{J}1(-, ). -()f,-(.. 4.3.1).,,..,( [102] rigid body re-sponse,..). [12]., . ,, , , .4.3.6. k-, ( ) 0 , ,) ( ) 1 (pk kN N F K (4.157)( ). - , (4.157) .,(4.157),- (.. ):) , , () ( ) 1 ( pik ik iN N F F K = , (4.158) Fi (4.89)(4.94). 4.4. 1294.4. , , (, , -).--,- .-. . 4.2 4.3. .,(,,).,.-, . 16.3 . , -,. , .,,- , . , , . 5 ( , ) -,-. -[29, 50]. .5.1. , - -,-.,,-. . . , , . , ,,- .,- (.,,[66]).,- - -.,- -.,- , , ;,,,- . , - . . . ,-.(-5.2. 131 , ) , .- , .--,- . , --,-.[29],,( ). , , .--- , . - , -,,- .5.2. -,-[39, 57,70].-.-,(),-., -,- . -.,-,- . ,, . -. 5. 132.,- - , (-),...-.- -, .. -,.--, , . - .5.3. 5.3.1. . . 5.1 - . l, (/). )P1(t)yPk(t)yxy0sk(x)p(t)ys1al)xyF(x) F(x)yyal. 5.1. : ; x = s1, , sk -P1(t), , Pk(t),,(x)p(t).-,p(t)(/),(x) .--. ,--( x = a)--M,--,-. -5.3. 133I,. 1.(),,M,- , -,[29].- , - .,,M,,(.5.1,). , ,, M. --y., ..- y - y (. 5.1,). , y, - , y . :==niiA10 , (5.1)Ai y (. 5.1). P1(t), ,Pk(t),(x)p(t)M,- . P1(t), , Pk(t)||.|

\|+ = = l s ai ia si ikiPi it P s la ls t Pay A ) ( ) ( ) (1 101 . (5.2) , - , .. 0 si a, - , .. a < si l. 01 1y dx x x la ldx x xat p Alaat p x ((

+ = ) ( ) ( ) ( ) () ( ) (. (5.3),,..(x) = 1, :. 5. 1342ylt p At p x ) () ( ) (= . (5.4) y Ma l ala lyayM AM ) ( =|.|

\|+ = .(5.5)(. 5.1,).x axy x F & & = ) ( ; a lx ly x F= & & ) ( , (5.6) axy x y = ) ( ,a lx ly x y= ) ( . (5.7) , =((

+ = laaFdx x F x y dx x F x y A ) ( ) ( ) ( ) (0 220223yly dx x la lydx xayylaa& && & & & =((

+ = ) () (.(5.8) (5.3)(5.8) (5.1), ) (t f yl=3& &, (5.9)+ + + = < l s ai ia si ii it P s la ls t Pa a l al Mt f ) ( ) ( ) () () (1 10((

+ + laadx x x la ldx x xat p ) ( ) ( ) ( ) ( 1 10.(5.10) (x) = 1) ( ) )( ( ) () () ( t pls l t Pa ls t Pa a l al Mt fl s ai ia si ii i21 10+ + + = < . (5.11) (5.9):0 00 = = y y ) ( ; 0 0 y y & & = ) ( , (5.12) t =0.-y0 = 0,.- 0y& ,5.3. 135 , .. t = 0. ,-y(x,t)(- . 3.2.2). =ldx x y T000) , ( & . (5.13),-,, . 4.1.2.. 0y& , axy x v0& = ) ( ;a lx ly x v=0& ) ( . (5.14) 202220022206 2yldx x la lydx xayTlaa&& & =||.|

\|+ = ) () (.(5.15) -lTy00 6= & . (5.16) (5.9) + =td fly t y00 3 ) ( ) ( & & ; (5.17)+ =td y y t y0 0 ) ( ) ( & . (5.18)y,max- tmax, , .. - 0= ) (maxt y& . (5.19) [29] . 1:3max , +yy y,(5.20) y .. 5. 1365.3.2. P1(t)0xyPk(t)(x)p(t)ysks1al. 5.2. MP1(t)xy Pk(t)(x)p(t)0sks1l. 5.3. R(t)ya(t) MM0xyP1(t)Pk(t)(x)p(t)R(t)lsks1a. 5.4. 0yMP1(t)(x)p(t)yxMMskPk(t)s1al. 5.5. , -..5.2.-,.- --.,--,-,.5.35.5. .1. --,-(.5.3).M.-,--,-.- -,--- M.5.3. 1372. ,(. 5.4).,,. M -R(t). ) () () ( ) ( ) ( ) ( ) ( t ya ld t p t P s la l a lMt Raa ll s ai ii& &3130 + + = < .(5.21) ((x) = 1)) () ( ) () ( ) ( ) ( ) ( t ya l a lt p t P s la l a lMt Ral s ai ii& &3 213 2+ + = 1x = 0.082. , ..5. 1685.5.6. - - . 5.5.5.m ,k F . 4.1.2 . 4.2. , . 5.4.3. m(5.98),(5.100).- (5.71):( ) , , , , 2 tg 4y x y xm b m a mbam R + +|.|

\|+ = . (5.101) , . 5.4.3.5.5.7. . 5.5.5. , , m,kF.4.1.2, . 4.2.. 5.4.4.-m(5.98).- (5.72): = =+ =41 ,41 , sin1cos1i iyi ixmm R . (5.102) , , - , ,) (t P F = .(5.103) , . 5.4.4.5.5.8. - , , . . 4.1.2.- . 4.2.,- .,-5.6. 169,,,- . . 5.5.55.5.6.,. 5.4.5.- (5.83) , 62Rm= ;(5.104),(5.76)(5.83),) ( 2 m m P R + = = .(5.105) , ,) (t P F = .(5.106) , . 5.4.5.5.6. , -,, . .,--. .6 - , -,-. -,. 4.- . -,,, . -,.- , .6.1. -, .P(t),tk,- , , =ktdt t P I0) ( , .(6.1) . mv q = ,(6.2) m ; v . ( ) , - :q q = I, (6.3) q , q . (6.2) 6.1. 171I v v m = ) ( , (6.4) v -, v . ,-,-,-.,--.-,-,-10 %(2025 %) .. 6.1,,--P(t), j, , k. -)ajP(t)xy0akP(t)v0kmkmnmjv0jm1)0v0(x)xyabl1f(x)p(t)l. 6.1. : , - ; - ajP(t),,akP(t),aj + + ak = 1. I. , ,,ajI,,akI.(6.4). , , - (6.4), v, v0:Imavjjj =0, , Imavkkk =0. (6.5) , , .. . , , -,..-- . , -:;j, , k(6.5),- .-,. 6.1,.x (x)(/).. 6. 172 p(t) (/) i(/).a x b- f(x) ( fmax = 1). ,xf (x) p (t)(/),-(..)i f (x)(/).,(6.5),- ) () () (xx ifx v=0, (6.6) . , ,.. f (x) 1, iv =0. (6.7) , .. . i x, -v0.(6.7) l1, (. . 6.1,):110lilv= . (6.8)-, l1. , ,,,,- . -., (x,y)(/2)i f (x,y)(/2) . ) , () , () , (y xy x ify x v=0. (6.9) , , -, , . .,, . 2 3.6.2. 173-,,- , .6.2. m k. I (. 6.2). - 2021mv T = .(6.10) , mImIm T2 2122== .(6.11)- x. 221kx U = . (6.12)myxkI. 6.2. , - mkIx = . (6.13)-A,- (2.8). x0 = 0; 0 0v x =&, 0vA = , (6.14) , /. , - (6.13): 02 2 2vmImImk mImkIx = = = = = . (6.15),,-n-,-. , -. 6. 174 1 (). - .6.3. 6.3.1. -. . 4.1, - . 5. - .- .6.3.2. a)Rx0.5lyyml --(. 6.3,). i+ = 400 . : l = 6 , b = 1.8,s = 0.25,J = 2.341034.E = 0.191011/2, = 900/. ( -, . 1, )y, 0 0.005 0.01 0.015R,0.050.1yRymax y. 6.3. - : -; ( )).--,-,M = 0.16106 .--.,--,--(.6.3,). -6.3. 175 . 4.2:my = 0.504l = 0.5049006 2722 , (6.16)= = =33 113 ,610 34 . 2 10 19 . 015 . 49 15 . 49lEJk 0.101108 /. (6.17)(- ). R , .. .-(5.100)M = = 0.16106 ,-. , , = = =610 16 . 0 446lMR 0.107106 .(6.18) y:0106 . 010 101 . 010 107 . 086 ,== =kRy . (6.19) . 6.3,. 9 . 60272210 101 . 08 , , ,== =mk/, (6.20) 1 . 09 . 602 2 ,= = =T. (6.21) , - 0.01 , .. 0.1T. . 6.1.2 - . -,. 6.2.i+= 400 .. 4.2,,-p,F = 0.64pl.,- , ,2765 8 1 6 400 64 0 64 0 = =+. . . lb i I .(6.22) (6.11):14042722 22765222= =mIT.(6.23). 6. 176-A, . - y (. 6.3,). (. 6.3, )||.|

\|+ =2yyR A . (6.24) (6.23) (6.24), 3610 8 . 720106 . 010 107 . 014042 = = =yRTy .(6.25)[29](. . 1):74 . 10106 . 010 8 . 7 0106 . 03 = +=+=yy yK . (6.26)K < 3, [29]- .i+-T . 6.3,, .. . ymax max2kTy = . (6.27)Rmax,-,. 6.3, Rmax = kymax. (6.28)RmaxF, ymax. . 4.2F- p Rmax = F = 0.64pl. (6.29), ymax lRp64 . 0max= . (6.30), .6.3. 1776.3.3. - , 1, , -i+ = 800 .,- : (2)M =(3)M = 0.2106 , (1)M == 0.16106 . : (1) = 0.31, (2) = (3) = 0.34. -a)R1x0.5lyym1lk1)Rx0.5lyym2lk2(2)M(3)M--[29](.. 1).- .---(. 6.4,)..4.2,- -. 6.4. : - ; :my = 0.406l = 0.4069006 2192 , (6.31)33 1131610 34 2 10 19 0 8 204 8 204 = =. . . .lEJk= 0.422108 /. (6.32) 75 138219210 422 08 ,11..== =mk/. (6.33) 045 075 1382 211..= = =T. (6.34). 6. 178 . 6.1.2, - ,0.1T. , ., -.-q 242) 1 (qlM = , (6.35) 122) 3 ( ) 2 (qlM M = = . (6.36),- -0 0.01 0.02 y, R, 00.10.2R1y2ymaxBRy1A. 6.5. .,, ..-,- (. 6.5). k1 OA . y1 A EJqly38454= . (6.37) (6.36) EJl My322 2) (= . (6.38), 23 112 62 (2)110 506 010 34 2 10 19 0 326 10 2 032 = = = .. ..EJl My. (6.39), ,6 2 81 1 110 214 0 10 506 0 10 422 0 = = =. . . y k R. (6.40)- : ,- (2)M (3)M( AB . 6.4,). -6.3. 179,..,, . 5.5.3. : -AB.-, (. 1):33 1132610 34 . 2 10 19 . 0 15 . 49 15 . 49 = =lEJk= 0.101108 /. (6.41) AB . 6.5.R()(5.107),,, a = l/2:=+ += + +=lM M Ma l aa l M M l MR) ( 2 4) () )( () 3 () 2 () 1 () 3 () 2 () 1 (66 610 24 . 0610 2 . 0 10 16 . 04 = + = .(6.42)y2AB :=+ =2 1 1 2kR Ry y286 6210 763 . 010 101 . 010 214 . 0 10 24 . 010 506 . 0 = + =. (6.43) . 6.5.,1.p-F = 0.533pl.- , ,4605 8 . 1 6 800 533 . 0 533 . 0 = =+lb i I .(6.44) (6.11), 48372192 24605222= =mIT.(6.45)ymax,,- . ymax T- ( . 6.5 ):. 6. 180) (2) )( (22 max 1 2 1 1 1y y Ry y R R y RT + ++ = .(6.46) =((

+ + =2) )( (211 2 1 1 12 maxy y R R y RTRy y

+ =210 506 . 0 10 214 . 0483710 24 . 0110 763 . 02 662(( + 210 506 0 10 763 0 10 24 0 10 214 02 2 6 6) . . )( . . ( = 0.023 . (6.47)- . 5.3.3. :008 06024 0 2 23 2..max) ( ) (== = =ly , (6.48) 016 0 22 1.) ( ) (= = . (6.49)[29](..1). 003 . 0035 . 0+ = .(6.50) 044 034 0003 0035 0003 0035 01 32......) () ( ) (= + = + = = ,(6.51) 045 031 0003 0035 0003 0035 01 1......) () (= + = + = .(6.52) , 044 0 008 0323 2. .) ( ) ( ) ( ) (= = < = = ;(6.53)045 0 016 011. .) ( ) (= < = .(6.54), .i+,T0Ay1. 6.5, , . , . 6.3.2.-ymax 0A:6.3. 1811 max2kTy = . (6.55) Rmax = k1ymax. (6.56),- . . 4.2, Rmax = F = 0.533pl. (6.57),,- ymax ,lRp533 . 0max= . (6.58),- .--(.. 5.23).. . 5.5.3, ,..,,-.-, , -,-,- .6.3.4. (-), . : a = 6 , b = 4 , h = 0.25 (. 6.6,). E = 0.21011 /2, = 0.2, = 525 /2. . i+ = 1200 .,--,:mx, =my, =0.12106 /.xy : x =y = 0.34. - [29] (. . 1).. 6. 182.-- . 4.2. :m = 0.25ab = 0.2552564 = 3150 .(6.59))a = 6 b = 4 xy)0 0.005 0.01 0.015 0.02 y, 0.51R,ymaxyR. 6.6. : ; (3.111) 823 112310 27 02 0 1 1225 0 10 2 01 12 = == .) . (. .) ( EhD .(6.60) =|.|

\|+ =|.|

\|+ =22 28 422 24416144 6 10 27 0 1 14. b aDabk 0.128109 /. (6.61) k - . ,(.. 5.5.5). . 5.23,, .. , k(- y . 6.6,). . 5.5.3, - . R -(.. 5.5.5). (. . 6.6,) :110 12 010 12 066== =..,,xymmk ;5 146. = = =ba . (6.62) (5.42) 84 01 1 5 1 3 15 13tg2 2.) . (.) (= + = +=k k k . (6.63)6.3. 183(5.111)( )7 6 10 112 0 84 0 5 1 10 12 0 4 tg 4 = + =|.|

\|+ = . . . ., ,y xmbam R . (6.64) 29710 875 010 128 010 112 0 == = ...kRy . (6.65) . 6.6,.,- (. . 4.2),52510 27 0416121 1282 2 2 2 1|.|

\|+ =|.|

\|+ = =. Db af f = 32.16 . (6.66) 031 0111. = =fT . (6.67) , T1,.. . 6.1.2 . . -. 4.2,-p (t),F = 0.405abp (t). , 11664 4 6 1200 405 0 405 0= = =+. . ab i I . (6.68) (6.11):215213150 211644222== =mIT. (6.69)-,A-. ymax. A ( . 6.12, ):||.|

\| =2max yy R A . (6.70) A = T, 0236 010 112 021521210 875 0272max...=+= + =RTyy. (6.71) - , (5.54),. 6. 1840236 0 41.max= = b y . (6.72)-,(5.57)(5.58). x0118 0212. = = x,(6.73) y001 0 84 0420236 0 tg22. . .max= = = byy.(6.74).- [29] (. . 1)003 0035 0.. + = .(6.75) x y , x , y , :044 034 0003 0035 0, ,.... = + = = y x.(6.76)(6.76)(6.73)(6.74),,.-, . T . 6.6,,.,. 6.3.26.3.3.-ymax- -:2 k T y =max. (6.77) Rmax = kymax. (6.78),Rmax-,. 4.2,- , ,abRp405 0.max= . (6.79),, .6.3. 1856.3.5. . 6.7,. 6.3.4.i+ = 2500 .-mx, my,, , mx, my,, ,:mx, = my, = mx, = my, == 0.12106 /.- : x =y = = 0.34.)a = 6 b = 4 xy)y, R,0 0.01 0.02123ymaxyR. 6.7. : ; ,-, , . .. 4.2,- m = 0.14ab = 0.1452564 = 1764 .(6.80)-, .. , . 6.3.4, D = 0.27108 . . 4.2, =||.|

\|++= 606 0723 02 24 4 4..b ab aabDk92 24 4 8 410 26 0 606 04 64 64 610 27 0 723 0 =||.|

\|++= . .. . /.(6.81) - (. 6.7,). ,. 6.2.3.. 5.4.3 5.5.6. -,(. 6.7,).- (5.74)(5.76) :. 6. 186110 12 010 12 066== =..,,xymmk ; 5 146. = = =ba ;110 12 010 12 066===..,,xxxmm ; 110 12 010 12 066===..,,yyymm ; (6.82)111=++=xy ; 110 12 010 12 066===..,,xymmk . (5.75):84 01 1 1 1 5 1 3 1 15 13tg2 2.) . (.) (= + = += k k k. (6.83)-(5.116):( ) =++|.|

\|+ = 2 tg 4, , , , y x y xm b m a mbam R ( )7 610 35 0 4 6 2 84 0464 10 12 0 =((

+ +|.|

\|+ = . . .. (6.84) 0135 010 26 010 35 097...== =kRy . (6.85), , .. ... 4.2,,p (t)F = 0.25abp (t).-, 15000 4 6 2500 25 0 25 0= = =+. . ab i I . (6.86) (6.11):637761764 215000222== =mIT. (6.87)-ymax.A(. 6.7,), , :||.|

\| =2max yy R A . (6.88)6.4. 187 ,025 010 35 06377620135 027max...=+ = + =RTyy. (6.89)ymax-,. 6.3.4. -, . . -.,. (6.116):025 044025 041. .max= = =by . (6.90) [29] (. . 1) 044 034 0003 0035 0003 0035 0...... = + = + = .(6.91) , 1 < , .. . T . 6.7,, -., ,abRp25 . 0max= . (6.92)6.4. -- , - . ,,-, . , -1 (0.1, , , 0.20.25 -).-, - ..-,- , . II 7 (),,,,-, . . -, (, ) -,, .(,.)().- , .:- ( ). -,.-, .. , ., . , - ,... .,-.,(..8). ..,,- , - 7.1. 189 , . ---:;,;- . .7.1. .XVII .,,.- , -(-).,h+h,.- , , , - . . ,,,- .,-, . . 7.2, . 11.-- , . 7.3 7.4.7.2. - - - - , -,,.- [53, 84].--.-.- . , - , , ,: . -. 7. 190 [53, 84], , .., l v0 -(. 7.1,),.,..,(. 7.1,).,.,, v0.)v0xl)c)c. 7.1. , - ,-(. 7.1,).--,-.- , -v0,..,.- v0.,,, . E c =.(7.1) E ; . -,- c l t =1, (7.2)- c l t 22 = . (7.3) [84]==K 5 3 12 202 2 21 8, ,sin cosilat ilx ilii cl v .(7.4)7.2. 191 (x = 0) ==K 5 3 1021 4, ,sinilat ii cv . (7.5) t = t1 cvcvi cvnii0 0110441 2) 1 ( 4= === . (7.6), E v E0= = .(7.7) ., , E A v A R0= = ,(7.8) . , .. 0 t t2. (7.6) - , , , -.,-- .,, . -, .(1908 .).- , ,.,,.- R - :2 3 K R = , (7.9) K , , .,... [7]- , . 7. 192 = R , (7.10),, - . .. - [53].,.v0-.(7.7). , , E0 . (7.11) h gh 20 = ,(7.12) E gh22 . (7.13), h . 20. = 1.25Ryn= 1.250.245109 = 0.3109 , Ryn , -[6];1.25,- [29]. - = 7800 /3;E = 2.11011 . (7.11) (7.13), v0 7.4 /, h 2.8 .25. = Rb = 0.18108 ,Rb, [29].- = 2100 /3; E = 3.21010 . (7.11) (7.13) v0 2.2 /, h 0.25 . , . - ()-[53].- . , -. , 7.2. 193.,,(,) , . [53] , l -.,,,, -, ,. , l, - .,-.. 8, . ,,,,,-. - ,,-.-,-,-, v0, 0. , -, v0 0 .- .,-,(,- ), , .- , -,,- .,,- - , -. . 11.1.2.. 7. 1947.3. 7.3.1. .-, :, , -,.-,:,,,., : , , - . .)txd)). 7.2. *: ; ; ,,- . - ( -)(.7.2,).-.,,. . , ,(.7.2,). , ..

* , . 7.2, -:- spalling; scabbing; perforation.7.3. 195t/d < 5,- .,-, (. 7.2,).7.3.2. ,(. . 17.2).,-,-,- ., , - . ,-. - , .(-),-.-.,-.-, ,( ), 150 /. - .,. , , -,.-,,,.- [99, 127, 135].-,[100],. :. 7. 196t ;tp , ;ts , - NDRC*[127,135].- (.. )0.31.5 %-. > +|.|

\||.|

\|=,;..2 10002 100048 108 10dxddVKNQdxdVKNQdx (7.14)**:x, ; d , ; Q , ; V0,/;N , ,=; 144 1 00 1 84 0 72 0.; .; .; .N (7.15)K , ,2 1180= ) ( f K , (7.16) cf ,/** ( , [29], 07 1 R fc. =). tp |.|

\| +=. . . .; . . . .35 1 718 0 19 35 13 35 1 24 1 32 12 dxdxdxdxdxdtp(7.17) ts

*NDRCNationalDefenceResearchCommittee ()** 1 (1) = 0.0254 ;1 = 12= 0.3048 ;1 = 4.445 ;1 /2 = 6890 .7.3. 197|.|

\| +=. . . .; . . . .65 0 06 5 91 775 11 65 0 36 1 12 22 dxdxdxdxdxdts(7.18),(7.14),-, 0.6 x/d 2.0. -,,tpts.-BechtelCorp.CEAEDF.,(7.14)(7.18)-,t/d 3, tp ts . -,-d0d. x (7.14), N = 0.72 d = d0. (7.30) (7.17), 220d d d = .CEAEDF*[127,135]. ( )dQV f tc p75 00375 0765 0...= , (7.19) (7.19)(7.21) , . , -:1.5 t/d 3;-3000 cf 4500/2(0.21108 cf 0.31108);155 /3(2500 /3);0.81.5 % .BehtlCorp.[18, 134, 135]:csf dV Qt=2 004 05 15... ; (7.20) . d0 :

*CEACommissariatlEnergieAtomique- (); EDF Elecricit de France.. 7. 198csf dV Qt=2 0004 042 5... . (7.21)s125 0 2 06 00. . d s .,, ts tp 1020 %.UKAEA* [103]. .UKAEA[103],-, , , . .1. - . , -,,-,,()- , -.- 2 13 222 1 6 13 0 3 1 ) . ( . +||.|

\|= rmp tf Vpcy c , (7.22)tp , ;m , ;p , ;cV ,, 45 < V < 300 /; , /3;fcy ,(:150 ,300 ),5106< fcy < < +< =2. 9395 02; 22 0 0605 0 25 00.22; 55 022 .. . ..G (7.29) = x/d; N , (7.15). , (7.28):.336 6/ 200000 5000; 10 44 10 22 /; 300 25< < < < < 0.75 50 +100 % < 0.75.30 /,- . - .. 7. 202.-, , - ..ts,- , 33 . 5 Gdts= , (7.31) G (7.29). : .326 6/ 40000 1500; 10 44 10 26 /; 238 29< < < < < 0.05.,( s/d: -s/d > 0.036;s/d > 0.068).,-, (7.28) , -,- . , (7.22),. , ,s/d < 0.05. , 20 %, s/d < 0.025.7. . (7.28), 2 42< = G , (7.33) (7.28) d. , - (7.34). - (7.33) 20 +60 %:7.3. 203. ; . .;32 06 6/ 850000 200000 08 0 03 0/; 170 26 10 36 10 22< < < =;) (;) ( ) ()] ( [2 12 2dt xdt xdt xt x f (7.39)x(t) , ; v(t) -,/; , (7.14). x(t) v(t) - ) ( ) ( x S t p x m = & & ,(7.40)m;S(x).-:x(0) = 0; 00 V x = ) ( & . R(t),, .7.3. 213,R,-.x :220mVRx = .(7.41) 202xmVR = .(7.42)R, 202xVmRa = = . (7.43)x,- ,02202 2VxVx+ = . (7.44) , :02Vx= . (7.45),, , R -,- , (7.52)(7.54) [118]. -, x. NDRC , 15 %.,, , --, , .-,-.- , 1 m mmm+= , (7.46). 7. 214 m ,=) (2) , (Sdxdy y x w m ;(7.47) ; w(x,y) ;S .,,,. 8 .7.4. -. -.,, . , . . BRL* [135]:2202 3000 120 1sKDVdt=, (7.48) t , , ; d , ; V0 , /; Ks ,(Ks1);D , /3,3dQD = , (7.49) Q , ..,-, , , . SRI* [127]:

2020452 . 01283DVdtdBdt=+, (7.50)

* BRL Ballistic Research Laboratory ( -, )*SRIStanfordResearchInstitute(-, )7.4. 215B ; .SRI- :2 D 12 /3 (0.55105 D 0.33106 /3);2 B 12 (50 B 300 ); 0.062 d 3.5 (1.6 d 89 );70 V0 400 / (21 V0 120 /);8 0 1 0 . . d t ; 8 5 d B ; 100 8 t B .,,-, B/d B/t, -.,B/d > 8B/t > 100,(7.50)- 2020452 . 01282DVdt Fdt=+, (7.51)FF = 100t/dF = b/d(-, F 8). , ,BRLSRI. B/d < 6BRL- , - SRI.-,SRI,25 %., [135].8 ,,- -(-).,,-,, () . -,. , , , . , . 7: . , -.,-:,..()., , .,-. .8.1. 8.1.1. , -., . . [129].- (x) -P(x)(.8.1).- , -(.,,[109,110]), . ,-8.1. 217 . ( -) .,(). . -(- -)*.-(. 8.2,):-- (1),01 = x& ,-(2),-) ( ) (2t t x &&= .- (1):)) P, 40302010x, 0 2 4 6 8 10 12 14 , /01000200030004000P. 8.1. - Phantom RF-4E [129]: ; P ( ) t d t t d t mt = =0 01) ( )] ( [ ) ( &, (8.1) (t) , (. 8.2,).- (2):m2(t) = mc m1(t), (8.2) mc .- , :dtdmu FdtQ di eiirrr+ = , (8.3)iQr i- (i = 1, 2);eiFr

* [19, 98] ,.,,,,, ..8. 218 )(t)Rm1(t) m2(t)x(2) (1)y0) (t &)R(1)P(x)xm1gy0. 8.2. - : (1) (2) ; , -;mi(t) -;ur-----,-(). (1) , , (8.3), OX (. 8.2,) 01 =xQ ; )] ( [ ) ( t P t R Feix = ; ) (t ux& = ; )] ( [ ) (1t tdtdm &= , (8.4) R(t) , P[(t)] , -.(8.3),,- [129]:)] ( [ ) ( )] ( [ ) (2t t t P t R &+ = . (8.5) , ., ,..0 ) ( t &, R(1)P() xgm1(). 8.3. , - - (1) - . ,(. 8.3).--,-(1)-, x.- , , - , - , OX8.1. 219( ) sin ) (1gm P R Feix = . (8.6),..0 ) (t &, sinceixgm F = . , = + +=. ) ( sin; ) ( )] ( [ ) ( sin )] ( [ )] ( [) (0 0 21t gmt t t t gm t Pt Rc && & (8.7), , - , R(t) :Rn(t) = R(t)cosR(t) = R(t)sin. >,-. . 8.2.3 8.2.4.8.1.2. -(8.5),) (t ) (t & [135]. -. .1. , . ,-,1, 2, -,(.8.4).- 1-l1 =0.-- 2xm2(t) m1(t)A1R(t)y0gmcA2AxAlA2(t)l. 8.4. xA = =) () () () (22 221 1SmdmllA, (8.8) m2() ;.8. 220 d ml= ) ( ) (2;(8.9)S2() , d Sl= ) ( ) (2. (8.10) | | ) ( ) ( 2 22 2 1 11m Sm ml m l mxc cA AA =+= . (8.11) 0x:) (t R x mA c= & & (8.12)( , ,).(8.12),(8.11):) ( ) ( 22& & && & + = m x mA c. (8.13)(8.2),(8.5)(8.13)(8.12), :) () (1m mPc =& &.(8.14) :0 0 = ) ( ;00 v = ) ( &. (8.15)(8.14),- , dddddtddddtd221& &&& && &= = = = (8.16) 2&= z . (8.17) 1- ) () (12m mPddzc = (8.18) 200 v z = ) ( .(8.19)8.1. 221 20012 vm md Pzc+ = ) () (. (8.20)= 01) () () (m md PFc, (8.21)202 v F z + = ) ( . (8.22) (8.17) (8.21) ) ( ) ( F v 220 =&. (8.23) t(), (t):= 0202 ) () (F vdt , (8.24) t(), (t). -)] ( [ t P ,)] ( [ t ,)] ( [ t &)] ( [ t m 1, (8.7) - R(t).max,.(8.23), 20012 2 v dm mPFc== max) () () (max . (8.25), = max-(8.24).,-.2. -,(.8.5).-,- 0y ( ) .(t)xAxAygmc0R(t). 8.5. .8. 222 0x: sin ) (c A cgm t R x m = & & . (8.26) (8.7) (8.14), : sin) () (gm mPc+ =1& &. (8.27),..(8.15).-, , (8.23) ) ( sin ) ( F g v 220 + =&, (8.28)F()(8.21). : += 0202 2 ) ( sin) (F v gdt . (8.29)max,-(8.28).,, .t(),(t)) (t &, (8.7), - .8.1.3. , , ..const1 = = P P ) ( ;const1 = = ) ( ; 1 1 = m ;l mc 1 = . (8.30), , . (8.21) |.|

\| ==lPld PF 111011ln ) ( . (8.31), 0.(8.30) (8.28), +|.|

\| += 0 2011122 vlPgdtln sin) ( . (8.32)8.1. 223max,- (8.28). (8.31) , max 0 11120=|.|

\| + +lPg vmaxmaxln sin .(8.33)max,-, . 1 P1, v0, max/l m1() .8.2.2. , - . , .-0xyz(.8.7).8.2. 231,..z() . --xkyk.-,- , ,,-.---: (. 8.7).ykyBxAxkxByAxRkygmcRkxgmkylABA00. 8.7. - mk,- , Rkx(xk) Rky(yk). , -, Rkx=kxxk, Rky=kyyk.B+-.,,A,lA (8.59). B:c kA c k kBm mx m x mx++= ; (8.84)c kA c k kBm my m y my++= . (8.85) : cos cos2 2ck A k Am m Sx l x x = = ; (8.86) sin sin2 2ck A k Am m Sy l y y = = . (8.87) -S2 = S2()m2 = m2().(8.86), (8.87) B:. 8. 232cos2 2c kk Bm mm Sx x+ = ; (8.88)sin2 2c kk Bm mm Sy y+ = . (8.89) + .- . :( ) sin sin ) (c k k kx B c kgm gm x R x m m + + = +& &; (8.90)( ) cos cos ) (c k k ky B c kgm gm y R y m m = +& &, (8.91) Bx& & By& & B. (8.88), (8.89) :((

+ = + cos ) (2 222c kk B c km mm Sxdtdx m m& &; (8.92)((

+ = + sin ) (2 222c kk B c km mm Sydtdy m m . (8.93)(8.92)(8.93)(8.90), (8.91), :+ = + + cos ) ( sin ) ( cos ) (22 2 2&& && && &S m m m m xc k k; sin ) ( ) ( sin 2 cos ) (2 2 22 c k k kxm m g x R m S m + + + +&&&(8.94)+ = + + + sin ) ( cos ) ( sin ) (22 2 2&& && && &S m m m m yc k k. cos ) ( ) ( cos 2 sin ) (2 2 22 c k k kym m g y R m S m + + +&&&(8.95) +-:eMdtK drr= . (8.96)Kr ; eMr - . eMr en n eF r Mrrr = , (8.97) enFr n- ; nrr - -. 0 j gm R i gm R Fk ky k kx er r r) cos ( ) sin (0 + + + =; (8.98)- 8.2. 233j y i x rk kr rr+ = 0. (8.99) A j gm i gm Fc c eAr r r + = cos sin ; (8.100)- j y i x rA A Ar rr+ = . (8.101) , | + + + = ) sin cos )( ( k k c k kx k ky k ey x m m g R y R x k Mr r| ) cos( ) (2 2 + m S g . (8.102)- n n nv m r Kr rr =, (8.103) nrr -n-;mn-; nvr.- ) (1 0 m m mk + =, (8.104)0. j y i x vk kr&r&r+ = 0. (8.105) xky&y &.ykxkCkx&&. 8.8. ) )]( ( [1 0 k k k k ky x x y m m k K& &r r + = . (8.106), - (. 8.8). d mC) ( = ; (8.107)- = + = j y i x rC C Cr rr] sin ) sin [( ] cos ) cos [( + + + =k ky j x ir r; (8.108)+ + + = ] sin ) sin cos [( & &&&rrk Cx i v] cos ) cos sin [( & &&&r + + +ky j. (8.109). 8. 234 { + + + = ) cos sin ( ) sin cos ( ) ( k k k k k k k k cy x x y y x x y k K d&& & & &r r+ + + + + + ) sin cos ( [ ) sin cos ( k k k ky x y x& & &2} d x yk k) ( )] sin cos (2& & & &+ + + . { + + = = ) sin cos ( ) [(2 k k k k k klx y y x x y k K d K& & & &r r r + + + +22] ) sin cos ( ) cos sin ( m y x y xk k k k & && }2 2I S x y y xk k k k & & & & &+ + + + + )] sin cos ( ) sin cos ( [ , (8.110) I2 ,= =ld I I ) ( ) (22 2. (8.111) + - (8.106) (8.110):). 2 ( ) cos sin ()] sin cos ( sin cos )[ () )( (2 2 2222 2I S m y x my x x y S mm m y x x y Kk kk k k kc k k k k k+ + ++ + + ++ + = &&& & && &r(8.112)(8.102)(8.112)(8.96).-,+ + + + + + ] cos ) ( ) [( ] sin ) ( ) [(2 2 2 2 S m x m m y S m y m m xk c k k k c k k& &+ + + + ) sin cos )( [( ) cos sin (2 2 2 k k k ky x S m y x m& && & + = + + ) cos sin )]( ( ) ( [ ] 222 222 222 k ky x S m I S m&& + + + + ) sin cos )( ( )] ( ) sin cos ( [ 22 2 2 k k c k k ky x m m g S m y x m&&kx k ky kR y R x m S g + + ) cos( ) (2 2 . (8.113) (8.94) yk, (8.95) xk,(8.113),- :=||.|

\| + + 2 22 2cos sinS mI Sy xk k & & & & & &) cos( ) ( ) sin cos )( ( 22 2 + + + = m S g y x m m gk k c k&&.(8.114)8.2. 235(8.94),(8.95)(8.114)-, : xk, yk, . --+., -) (~1 m m mk + = .-Rkx RkyP() T (. 8.9). - ,RkxTyRkyxgmkgm1P(). 8.9. , , . , - gmk gm1(). , ( )( ) ,~~;~~ey y kkex x kkFdtm du ydty dmFdtm du xdtx dm= += +&&&&(8.115) kx&, ky& ; ux, uy -, : cos&& + =k xx u ; sin&& + =k yy u ; (8.116)exF , eyF , + :( )( ) . cos ) ( cos sin; sin ) ( sin cos11 m m g T P R Fm m g T P R Fk kyeyk kxex+ + + + =+ + + =(8.117) (8.59) (8.60) (8.58), : + + = + cos )] ( [ ) ( ) (21&& &P x R x m mk kx k k sin ) ( sin1m m g Tk + + ; (8.118)+ + + = + sin )] ( [ ) ( ) (21&& &P y R y m mk ky k k cos ) ( cos1m m g Tk + + + . (8.119) (8.94), (8.95), (8.114), (8.118) (8.119) -. 8. 236: xk, yk, , T. -:xk(0)=0; 0 ) 0 ( =kx&;yk(0) = 0; 0 ) 0 ( =ky&; (0)=0; 0) 0 ( v = &;RyRkxxRkygmkRxy0. 8.10. , - (0) = 0; 0) 0 ( & &= ; T(0) = 0. ,- . -,-RxRy(. 8.10): sink kx k k xgm R x m R + =& &; (8.120) cosk ky k k ygm R y m R + =& &. (8.121)8.2.3. 0 > , - . --[131]. [8, 9]. ,,. , , , , ,.- - x y (. 8.11).-,,-.+-(8.49), xc , ( )( ) c kcc kcm m mm mS my y++++ = sin2 2. (8.122)-(8.94),(8.95) cm & &:( ) ( ) = + + + +c c km S m m m m y & & & && && &cos sin2 2 2( ) ( ) ( ). cos sin sin y R m S mky + = && &&222 222 (8.123) + ,- :8.2. 237+ + +||.|

\|+ coscoscos sin2 222 2& && && & & &S mmyS mmxc( ) =||.|

\| + + coscossin22 222 22 2&&& &S m S mI S||.|

\| + +||.|

\|+ = sin ) cos(sin1 22 2 2 22S mmgS mmc&&. (8.124). 8.11. : + ; , ; , xfRRTPy0gm1)RRkygmkRkxfRxy0)m1(t)m2(t)xxxylRkyyymkRkx0a)(8.49),(8.123)(8.124)-,:x,y,. (. 8.11,). -R,, f R, f ,f = tg. (8.125). 8. 238 :( ) sink kx kgm x R R x m + =& &; (8.126)( ) cosk ky kgm y R fR y m + =& &. (8.127) :( ) ( )) sin (cos f gmy R x fRx f ykky kx + =& & & &; (8.128)( ) sink kx kgm x R x m R + =& &. (8.129),,(. 8.11,).-,,-, := + + sin sin cos& & & & & &1 1 1m y m x m( ) | |+ + + = sin ) ( cos& & && &y x2( ) ) sin( sin cos + + + +1gm f R P ; (8.130)+ + + = cos cos sin1 1 1m m y m x T& & & & & &| | + + + sin ) ( cos& & &&y x( ) cos sin ) cos( f R gm + 1.(8.131) (8.94), (8.123), (8.124), (8.128), (8.129), (8.130) (8.131),:x,y, ,,,RT.,,., , :(0) = 0; 0) 0 ( v = &; R(0) = 0.,, . [32] - , .8.2.4. , , - . , - , . , , (. .14.2).8.3. 239- , -.-. 8.2.3,.- -,(. 8.12).-,, [8, 98].RTrfRP. 8.12. 8.3. 8.3.1. -:, . ( ) . [86],,- , . (.17).,-,.(8.5), , .,,,- .- , -, - . -, [112].. 8. 240 , ,(),-.,-,/,--,.-,,.,-, [18].,-., . ,- , 7. . 12.8.3.2. LearJet-23. (. 8.13) [62] [112]. 57 , .. 8.13. Lear Jet-2313.8 ,3.84 ,10.85 .--5670 .-1 ,0.45 180 ..8.14,.-v = 360/.[62],S = 12 2(- l = 10 h = 1.2 ). , - 5 , -045,5 80 .8.3. 241) t, 0 0.05 0.1 R, 15105) f, . k0 20 40 60 80 1000.511.5 = 0.050.070.1. 8.14. Lear Jet-23: [62]; .8.14,k. - (.. -- ), f,,.8.14,, (.. 12).,k, 10 30 , - R(t). k - 1.21.3, .. - .k,. 4. ,R(t), . - k, .Cessna-210. (.8.15)[62][112].46.:8.6 ,9.9 , 11.5 . 1725 ..8.16,.-v = 310360/. S = 4 2 ( -l=8 h=0.5[62]).- , Lear Jet. -. 8. 242. 8.15. Cessna-210-.8.16,.--200 ,-.0.5 2. - .) t, R, 024680 0.01 0.02). . K00.511.52 f, 0 100 200 300 400 500 = 0.050.070.1. 8.16. Cessna-210: ; . .8.17;S = 2 2; 0 90 .) t, 00.01 0.02 R, 00.52.52.01.51.0). . k f, 0 200 40021 = 0.05 0.07 0.1. 8.17. : ; 8.3. 2438.3.3. -PhantomRF-4E. [128] - [112]. . 8.18,, -- . 8.1.--,- ,. 8.18. -Phantom RF-4E . 8.19,.) t, R, 0 0.02 0.08 0.04 0.061208040) R, t, 0 0.002 0.004 0.00610203040). . k f, 0 20 40 60 80100010.51.52 = 0.050.070.1. 8.19. -Phantom RF-4E: ; ; . 8. 244.8.20. Sandia National Laboratories () [133]-,. 8.19,,,, ,-.8.1,.20000 ,-215 /[110].0