ISBN , 2015
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- Marcus Bruce
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1 ... 5
2 : : /.... : ISBN ISBN
3
4
5 F F
6
7 E A
8 U
9
10 «-» ( 3.) (.3. ) » ( ).
11 ) ().
12 ( -.)
13
14 -. -.) [ ] [ 6] [ ] [] [3 5] [ ]. - [75 83]. - - :
15 - - [ ] [ ]
16 [ ] [8 3] - [ 33 35]
17 [ ]. - [ ]. 7
18 [ ]. -.. [99 c. ] : « ». 8
19 . x y z x y z [].. -. : f ( x y z... x y z... x y z x y z... x y z... x y z t) n n n f ( x y z... x y z... xn yn zn x y z... x y z... x n y n z n t). (.) n n n 9
20 . [55] x y z ) O- ( A A A l. - : xa ya za l. (.) l O -... ) ) O y O y x z l A x z l A A... :
21 . - Z Asin t. O ) ) A O A x( t ) z m x( t ) m z z z... : A x m. c x m A z... - : Asin( t) x. (.3) -. - []
22 [].. I 5 II 4 III 3 3 III 3 3 IV 4 IV 4 V 5 V 5 V 5
23 []. - ) B a ) ) A 3 U D B ) ) A U U A l B V V l B l V l V..3. : D
24 U
25 [] : [ ] [ ]. 5
26 - [46] [78]
27 [ ] y N J x Asint m F O P [ 68 78]
28 ) V c ) V n n c..5. : ) O ) x O x G G M P cos( t ) M P cos( t )..6. :
29 [ ] [64 65]. - ; [7 6] [3 4 79]. l l x c d/ d/ m c c M x..7. 9
30 [3]..8. x ) ) x c m m c M c M x x..8. :. - 3
31 [4] ) [4]. ) ) m c m g m m m m m m g..9. 3
32 [4] ) )... :
33 - [8 66] ) - ) [38]. 33
34 ... : (..)
35 [ 44]. - - [7]
36 -. - [34]. - [] [38]
37 []. 37
38 [3 47]. [49]. [79 8]. [35]. - - [ ].. - [ ] - I. II [ 48 49]
39 [98 3] [7] q q... s II. - II T - Q... Q - i s : 39
40 d T T Q dt q q... d T T Qi. dt q i qi... d T T Qs dt q s qs (.4) ( t)... q ( ) q s t II - - [7] [47]. [94]
41 . II -. - ANSYS Rigid Dynamics -. ADAMS []. -. [93]. -. [8] -. [54] [88] ;. - 4
42 -. [7] [85] n ) ) ) ) ) ) 4
43 . ) ) ) ) ) m m m m m m m m m m m m x
44 - [8 34] [4]. -. : -. - [4] :...3 [34]... 44
45
46 .4 [5]
47 -. [5] : ) ; ) ; 3) ; 4). : ) ; ) c ; 3) - ; 4) -. 47
48 .3... ( - ) [ ]. - - [ 4].. [4]
49 [4].. - : -..3 [4]..3 ) ) ) ) ) )
50 .3 ) ) ) ( ) (..) - (- 5
51 ). - - [4]. -. ( )
52 : - ; ; [4].. 5
53 [34] - 53
54 : ) m c b 6 ; ) 6 m r- - : F a m r F Fa cost. (.5) [4] : x a m r h ( m m ) ( ) 4h b. (.6) ( m m ) m m c xa x a max m r ( m m ) h h. (.7) h 54
55 .6..6 m r x a. (.8) m m
56 : : ) - ; ; ) - ;. 56
57 [39 5] [3 45] [99 46]
58 [ ] [ ]
59 [ ]. 59
60
61 ( - ; ). - [4 3 6]. (..7) :
62 [4 74 6]. (..8) [4]. 6
63 [5 7 3]. (..9)
64 : - ( - ); ( ) ; - ; ; [69]. - ;. [ ]
65 .. : (..) :
66 [4 9 3]. - [46] [ ].. -. Oxz - (..) - : m x mgsin (.9) m ; ; F fn Sign x ( x ) ; f ; N - ;. F 66
67 ... : N mgos (.) z N. : F f N (.) x f. - : T v v(t )dt T
68 - (.9). - t j [3] ma Sin t A. - ( 3 4) t j 68
69 t j. f f - []. - [7 48 8] (-..). - [3 5 6 ] [ ] [ ]. -. [ 54].. [ ].. 69
70 [ ]; - [36 ]. - (..): mx mgsin F my F y (.) F v f N v ; v v ; v N mg os z.... (.): - v F fn. 7
71 (.) -. - (.) -. - [66] r. (.) : r mos( t ) r msin( t ). (.3) x y z gf r (f ); gf r (.) (.3) : g f x r z Sin( t ) y r z os( t ) z r r r z ; arcosz. -. : [6]; [4]; ; [3]. [5 6 ] 7
72 [77] [ 64]. - 7
73 (). [6 53]; - [ ]. - : - ; ; [34] :
74 -. [55] [5].. - [87] [87]
75 - ( -.). - () [9 57]. - - s ; s. P r ; P r - ; : - - [ ]. - [4]
76 .. [86]: df f PP v ds v (.4) ( ) df ds ; f ; P P - ds : - ; v ds. (.4) : - ; - ( ). : - = ;. - ; - [ ] - []. 76
77 [4] - [36]. -. [49] (- ): c x A Sin t gf x. (.5) m c = (.5) -. 77
78 ..3. : = = 4 - ; = = [ 4 4] - : -. (..4)
79 ..4. G N N F F. - : mx mgossin Sin( )Sin N my mgsin os( ) F N J [ F os( ) N Sin( ) fn N fq fn u ]H / os (.6) ; ; u n q - -. (.6). - : -. 79
80 [67] N. - M fen Sin =. e=o ;. - : -.. t l - : J zt t efn ( tg( ) ln ) tg( ) l v t ; J z ; v -. 8
81 tg( p J ) ztv l ln. efn ( ) tg( )
82
83 - :
84
85 . ADAMS []. MS Software Inc. : (: 8..4).. Harris. Shoc and Vibration Handboo /. Harris.E.. New Yor : McGraw Hill Boo. 457 p. 3. Jianlian heng Hui Xu. Inner mass impact damper for attenuating structure vibration International Journal of Solids and Structures Vol. 43 Issue 7 August 6 Pages ISSN :. : // /.. // /.. // : - /
86 9.. - /.. // : /. -.. : :.. : / // (48) () /... : : c. 6.. /. // /..... : : /.... : /..... :
87 .. /.... : (. ) /... : /... : / : /.... // // I /....: : : /... // / :
88 3.. //. - /.... : :.. : // /... : /. -.. : /... : :. 6.. : /..... : : 6 /. :..).. : / : 6 /. :..).. : /
89 4. : 6 /. :..).. : / : / : : /... : /. // II /. // : /... : : - / /.... :
90 5.. - /.. // : :. : :. : /.... : : /... : /. // /..... : /..... : : : 9
91 : /... : /.... : /.. //. : /.. // : : I /.... : // /... :
92 7.. /. // : - /... : /... // /... // /... : /... : /. -. // /.... : /... : /. // :
93 - :..... : : /. -.. : c /.. : : : 5.. / /.... : : /.... : : : :. : : /... :
94 93.. /. // :.. - /.... : :.. I. /.... : :..... :..6 / : /.. // /.. // :. : /... : /... : :
95 3.. - //..: (6) :. : : : /..... : : : - /..... : : - /... : /... : c... /.... :
96 3.. /.. // : /... - : : : /. -. : :.. /. -.. : : /. -.. : : / : : :..6 / /.. :
97 5....: () /.. // : -. : : / : : /... : / : : /.. -. : : 97
98 :.. : 5.4. / : /.... : /.. // : / :. : II :. : II : -. /.. -. : // /... :
99 45....: /.. : /.... : /. // : /. - // : III : :..6 / :.. : : :..6 / : /. // - 99
100 .. : :.... : / : /..... : // [] : /.. : c /. -.. : :. /. -.. [.].. : /... ; : : // :..... :
101 / : - -. : /.... //. : : -.. : / :
102 ( ) [ 3 4].
103 .. H m -. Q. H ( t) Asin( t). t m H A m X(t). - A H ( t) Asin( t). t. : 3
104 X ( t) g t t X ( t) H ( t). X ( t ) ( ) H t (.) - : - (.) - X ( t) Asin( t) X: A = = 4 /c X H ( t t) t. - 4
105 (.) - t. t XH - H t : t 5n n =.. A = 4/c n. X H t t ) - ( (.) t t H ( t) Asin( t) H(t) t. - (.) : 5
106 X H ( t t ) g t t t X H ( t t ) Acos( t ) t tt X H ( t t ) Asin( t ) t t. (.) X H ( t t ) t t (.): X H ( t t ) Asin( t ) A( t t ) cos( t ) g( t ). (.3) t X(t) (.3) t H( t) X( t). m Q N. : mx N Q. (.4) N: 6
107 A g. A N ( t ) mg sin( t ) g (.5) - : N H ( t) mh Q. (.6) N H (t) : A = X H t t ). t ( X(t) X H ( t t ) H(t)... t - A :. X t ) Asin( ); ( t 7
108 . X ( t) A sin( t) t t t ). ( t A :. X t ) Asin( ) t t t ]; ( t (. X ( t) Asin( t) t t t ). ( 3. t - A - :. X t ) Asin( ) t t t ); ( t. X t ) Asin( ); ( t ( 3. X ( t) Asin( t) t t t ). ( -. R(t)=X(t) H(t). -.. []. 4 (). 8
109 9 ) ( t t X H ) (t H ) ( t t t t ) ( ) ( ) ( t H t t X t t R H H. :. ) ( ) )cos( ( ) sin( ) sin( ) ( t t g t t t A t A t A t t R H -.. t :.. ) ( : i t t N d t t i H i i (.7) i d.. ) ( t t R H t t t :. ) ( ) (! ) ( i t t i H i i H t t t t t R i t t R (.8) - H(t) :. ) ( ) ( ) ( ) ( ) ( t H t t X t H t t t X t t Q t t t X m t t H t t H H (.9) :
110 . ) ( ) ( ) ( ) ( ) ( ) ( t t t t H t t H H H t t H t t t X t t t R t H t t X t t R (.) ) (t N H : ) ( ) ( t mh Q t N H. (.) : m Q t t t X m Q t N t H H H ) ( ) ( ) ( (.) :. ) ( ) ( ) ( ) ( m t N t H t t t X t t t R H t t H t t H (.3) (.) n Q : ) ( ) ( n t t N m t t H n H n n n. (.4) ) ( ) ( ) ( t H t t X t t R H H :. ) ( ) ( )! ( ) ( ) ( i i t t i H i H t t t t N i m t t R (.5) i (.5) - : ) ( tt H t t N. (.6) : ] ) [( ) ( ) ( )! ( ) ( ) ( ) ( t t t t o t t t t N m t t R t t H H..(.7) ) ( ) ( t t :
111 R ( t t H ( t t) NH ( t) ( ) ) m ( )! t tt () t t. (.8) RH ( t t) t ( t ( ) t ). ( t t ) t R H ( t t ).. t - R H ( t t ) t t t ). - ( d i.. d i i. ( t t ) R H t t t ).. ( N N t ) N ( ). H H ( H t... A t : ) X t ) Asin( ); ( t g ) sin( t ) ; A ) t [ ) t...
112 : ) N H ( t ) ) N t ) N ( t ) H ( H. t - - : A.. g. t ). N H (... t - N : i RH ( t t d i : i t ) t t i.. - d d... d d. (.9). : d d... d d. (.). t (.9) - - [4] :
113 R H i RH ( t t) i ( t t) ( t t) o[( t t) i i! t i t t ]. (.) (.) (.) - -. R H ( t t ) t t t ). - ( t. i d i t.. : N d d... d d. (.) : R H RH ( t t) i ( t t) ( t t) o[( t t) i i! t tt i i ]. (.3) R H ( t t ( t t ) RH ( t t) o )! t t t (). (.4) R H ( t t ( t t ) )! R H t ( t t ) tt t t (.5) t t ) RH ( t t) t ( ( t t ) R H ( t t ). t.. 3
114 t - - t t ) R H ( A sin( t) g 3 A sin( t) g cos( t) >
115 ..4. : A = = \ T T T ] T [ A g A ; g A A T g g - 5
116 \ S ( t ) S ( t ) S max( t ) max ( t ) Asin( t S Asin( t. g ( arcsin( )) ( ) g arcsin( ) A A t g arcsin( ( A S( t ) ) ) g Acos( arcsin( A () ( A cos( t )) ) g ) [) g A arcsin( g ) t A ( ) )) ( ) g ( t ) S g A g Smax D A A sin( t ) g cos( t ) g A A g : A = = min max t t
117 [7 3] t t. : 7
118 S( t) t t S( t) (.6) t t
119 9.3 \ - N K }.. { max g A N K N - A g A g. arccos A g. arcsin. arccos A g t. arcsin t ) ( ) ( t t g t t g g A S N K }.. {.. ) ( ) ( ) ( t t g t t g g t S. max g A g A S. max g S A S D max. A g A g A g D. D
120 ..7. : ( = ) ( = ) A g A. - - : A g. (.7).8.
121 ..8.- : = 3 = 6 d = = = 3 - a b c; a b. K = A = = 84. = =. - ( - ).
122 ... : A = g 4 o = A o 4 = o 6... (.. = 6)
123 ... : H H H = 8 = = 37 A = h h h g h (a c) H c : A = S H h S H h
124 ..3. : A = A = A = 3 A = 4 A = 5 A = : A = A = A = 3 A = 4 A =
125 : S A = 5 = =
126 b. 3 (a c)...6. : b (a c) 3.7 (- )
127 ..7. : = = 9 = /...8. : 3 a e 5 I V
128 : ()
129 m H H( t) Asin( t) Q F. F Q. F a-.. b- N - F Q.... Q F : a c b c. 9
130 . S -. - H S : ms N Q F. (.8) F Q F
131 \ ) cos( ) ( ) sin( t mg f mg t ma ) ( mg f g A ) ( arcsin A mg f g t ) )( ( ) )cos( ( ) sin( ) ( t t mg f g t t t A t A t S ) ( )) cos( ( ) sin( max mg f g t A t A S ) ( ) ( ) ( mg f g A A t R ) ( mg f g A T
132 m F : T m Af ( mg f ). (.9) f (.9). f t max -. F.5. 3
133 f f -. f i..5 f t t F t t f t t mg t f max max ; max ; mg ma g arcsin A X - - X ( t) g ( t t A cos( - tmax t g X ( t) Asin( t ) A( t t )cos( t ) g ( t t ) T T ( t max ) g A g A g max wt ) ) g A g A g 33
134 T A g f f mg mg F -.5 f f. f : f. mg : (.3) A f g( ) mg. (.3) - : mg f mg ma. (.3) 34
135 .3.5. F. f f f f : f mg A T. (.33) f f g( ) mg mg - - : mg f mg ma. (.34) A g. A g - : g. (.35) g A. -. : mg f mg ma. (.36) 35
136 .. : mg mg f. (.37) f m( A ) - g. - f : mg m( A g).. (.38) : A = = m =
137 A. T - : f mg f mg A T. f g( ) mg (.39) - : : f mg f mg T. A f g( ) mg (.4) f mg ma f mg ma. (.4) T f A. m. 37
138 f f YOX. - F = ( f f ) - f f. -. E FD : ( mg mg) F ( mg ma mg).3 EFD. E F -. FD. - F FD - E F FD. 38
139 -. - DYO OB..3 II. - I. - AO ODX. AOYE BOXD. III. PQ. P Q.. - F Q. mg mg ma. PQ [5]. 39
140 .4. H m -. m - Q F p - -. p. J ( t) B sin( t). J H =. - H ( t) Asin( t) - t..4 - : - b...4. m H () (b) J: N Q F p V X: F p p( x J ). (.4) 4
141 t m H H. -. A m m = A = 96 = p = X(t). A H( t) Asin( t ). t. 4
142 X ( t ) X ( t ) - H t ) H t ). - ( ( : mx ( t) p( X J ) mg t t X ( t ) H ( t ). X ( t ) ( ) H t (.43) t - - : - (.43) - X ( t) Asin( t)..6 X H J. () () () mh ( t ) ph ( t ) gm pj ( t ) 3 () () () mh ( t ) ph ( t ) gm pj ( t ). (3) () () mh ( t ) ph ( t ) pj ( t ).6. - p : A = 4 = m = p = 59. 4
143 ..6. X II - - A : J ( t) Asin( t). -.6 : mh mh () (3) ( t ( t ) gm. ) (.44) 43
144 H ( t) Asin( t) : m Asin( t) gm. (.45) 3 m Acos( t). [] / ( ) J ( t) Asin( t) H ( t) Asin( t) J ( t) B sin( t) H ( t) m Asin( t ) gm gm pb cos( t ) 3 m Acos( t ) pb sin( t ) g gm t arcsin t arccos A pb gm pb. - - J =
145 45.8 ) ( t J ) sin( ) ( t A t H ) sin( ) cos( ) cos( ) sin( t m p t A g t m p t ) ( ) ( ) ( exp ) ( ) ( t H t t p gm t t m p t H p gm p m t X arcsin arcsin m p m p m p A g t ) ( ln max t ph gm gm p m t t ) ( ln ) ( ) ( max t ph gm gm p m g t H p m t H X 3 exp exp p mg m p g m p m p g A
146 .9. g A.9 g I p [ ) A g t p ( ) A g II A g p m A \ p p m p g A g A g t arcsin A - g A - A t arcsin t t g.7 p. 46
147 ..7. p: A = 89 I II a g A b P p p: = 6 A = 5 B = 86 47
148 .8 - B = 86. p A A. X A : X ( ) A. X ( ) A (.46). - - A. X ( ) A W- [] - p > p. 48
149 ..9. : a b c = 6 A = = X m = 3. p : H X 3 H m = = 6 p = A = 38 49
150 .4.3. [] (p = ). -. p - - p p ). ( p H(t) t p ( ) ; : X ( t X ( t p p ) H ( t ) H ( t p p ). ) (.47) A p t X t p ) A X t p ) A. t p p ( ( : p 6 p 8 m 6 A 6 A 8 5
151 - A - : t t p p ;. ( ) ; ( ) ( ); t p t p (.48). p : ( p) ( p). (.49) p p..3 p p ; p. - - (.49)...3. : = 5 m = = 6 p p. p; p - p p - 5
152 . p p ; p..33 p. p : I III p II p > : = 6 m = = 6 p = 4 5
153 [6 8 9] :
154
155 /.. // W-- : / : 6. 6 c /. // - 55
156 :..-.. : / // - : XVII. I. - : /. // // /... // /.. // /.. // / 56
157 .. // / /.. // (35) /.. // (6) : /. ;
158 m m l m m Z : P i Q i F i N N a b Z A sin t. 58
159 -. ai X i i mi m i. d d. P i - : : i Qi F i - P p X Qi mi g Fi fi i. (3.) i i X d X : X. (3.) d X. (3.3) N
160 :. - - : R Q Q F F N N. (3.4) X : R ( X ) X l. (3.5) X : N N N N. (3.6) X : ( X Q F X N ) Q F N. (3.7) : X X X X Q d F Q F. (3.8) 6
161 : - N N. (3.9) Q F N N N. (3.) X : X X X X X X X. (3.) (3.3) : X X. (3.) N N R Q F i i Pi i : R Q F P N Q F P N. (3.3) 6
162 m : R Z ( X ( X )) P p X X m. (3.4) m - : : px P m X. (3.5) R R R f z (3.6) ( X X R X ) R f z Z. (3.3) : R Q R F f P z Q F N P N. N N (3.7) (3.4) - : R P f z P N. N (3.8) : p X P X m P p X mx. (3.9) (3.9) - (3.8) : X p X f z m X p X N m X. N (3.) 6
163 : X ) X () X ) X (). (3.) ( ( N : m X p X m X px X () X X () X X fz N () (). N (3.) (3.). (3.) : m s ms X psx X f z X p sx N. N (3.3) X X f z N s - X X f N : z st X ( s) X ( t) e dt. (3.4) []. N : (3.3) X X X X f z N ms ps N. m s p s (3.5) 63
164 - (3.) : X X. (3.6) (3.6) : m s f z N p s m (3.7): s) N. (3.7) s p s L ( m s p s L s) m s p. (3.8) ( s : N L ( s) L( s) fz. (3.9) L ( s) L ( s) L ( s) f z Z ( Z Z ) - Z - W Z N N : L( s) ( s). (3.3) L ( s) L ( s)
165 .. W Z N ( s) : L ( j) A( ) (3.3) L ( j ) L ( j ) j -. L ( ) L ( ) s j: s s L L ( j ) m ( j ) m jp jp. (3.3) : (3.3) A( ) m jp. (3.33) ( m m ) j( p p) A ) : ( A( ) ( m ) ( p ). (3.34) ( ( m m) ) (( p p) ) : max N ( t) A A( ) (3.35) t (3.35). (3.35) - 65
166 : N A A( ) N. (3.36) (3.36) (3.35) : N ( t) N t. (3.37) : N N N t. (3.38) N X 96 N m i l i - i - p i F i 66
167 A 3 4. (. 3.. : 3 4 A 4 5 A ) (. 3. 5) : I II G N A A( ) D 67
168 (. 3. 4) (. 3. 5) I. A A A( ) N : 68
169 ( ( m ( m ) m ) ) ( p ) (( p p ) ) A m g f c. (3.39) (3.39) - : A m p m g (3.4) A- A : A A. (3.4) ( ( m m ) (( p p ) ) ) A A - A (3.4) : : A m p m g. (3.4) A g m gf m A p f. (3.43) c c (3.43) m - f c f c 69
170 ( ) -. (3.43) : D A g f p A 4 c A p g A f c p. (3.44) - : A g m gf m A p f. (3.45) c c D4 A f c p -. D
171 : D 4. (3.46) (. 3.4) - : D f 4 p c A p g A. (3.47) -.. ; -.. 7
172 (. 3.4) (. 3.4) () () D f f f c c 3. > > < () ( m m ) > = = = > < > = < > = = ( ) m ) ( m ( ) 3 m m ) c ( ( ) > < > () ( ) ( m ) m ) ( m ) ( 7
173 f - 5. f 6. f f c m g () f m ) : c ( m c c c gf D4 m c gf D4 m. (3.48) c 6. : f. A 4. c () 3.4. ) ( A () :. A g m gf m A p f. (3.49) c c 3.5 ( ) - 73
174 : m m m 6 m 6. 3 m ) - ( m.. () () : m m -. - (). - 74
175 () -. - m ). ( m [ 3 5] (3.39) p p fc : A ( m m m ) gm. (3.5) : A g. (3.5)
176 -. : (3.5) - A m gm (3.5)» «m m -..»
177 : 3 A ) ( A :
178
179 [ 6] s s - s 3.9. s H s. H - q [.. s] - q q qq qq lqq l qq
180 z X X H d a ) ) l m m F b Fg R F N N m m X d 3 l 3 F b Fg R 3 F X s s d ss l s s ss s m s s s F bn Fg n R n n F n n m n X s d s s s s s s lss s m s N n n Nn n m n z s s H s F b n F g n R n n F n n s s : ) : i i Z Z s m q m q q ) : F b i qq F i F g i R q q R qq i i.. n N qq N q q s q.. s i.. n n s d i. q n s i.. n.- q q-. -. i X i OX O 8
181 .. t -. t Z H H s A sin t Z n A sin t H Hn. -. b i F : : F F g i b i bi X i Fg i mi g F i fi F i i.. n. (3.53) X d X q... s. (3.54) q q q - : X d X q... s. (3.55) q q q Xq dq Xq q q-. (3.55)
182 - s Nq q q... s. N. q q
183 [ 4] ( z(t) N(t))
184 : R F N R3 F N R3 F3 N34.. Rq q Fq Nqq R qqfq N qq Rqq Fq Nqq.. Rn n Fn Nn n3 Rn n Fn Nn n Rn nfn Nn n. (3.56) n - : R X R3 3( X X3) R3 3( X X3 ).. Rq q qq ( XqXq ) R qq qq( Xq Xq ) Rqq qq( Xq Xq ) (3.57)... Rn n n n( Xn Xn ) Rn n n n( XnXn ) Rnn nn Xn. 84
185 : X i X i - X i i.. n. (3.58) X i X i X i - : X X X... X X... X X X 3 q n H n H H q n H H H H l 3 3 H n l d qq l d 3 3 n n n n n n l qq qq d l d. q n n n (3.59) Hqq Hq q H q( H Hn). s (3.57) (3.56) - : X F N 3( X X3 ) F N 3( X X3 ) F3 N34.. qq ( Xq Xq ) Fq Nqq qq( Xq Xq Fq N qq qq( Xq Xq ) F q Nq q.. n n( Xn Xn ) Fn Nn n3 n n( Xn Xn ) Fn Nn n nn Xn Fn Nnn. (3.6) 85
186 - q N : qq N N q q - N q q N q N qq q N q... s. (3.6) (3.3) : X X X... X... X X X 3 5 X X X q n5 n3 n 4 6 X X X X X X X q n4 n n 4 6 X X X X d d d n 4 6 q n4 n d X X X d 3 5 q d d n X X n4 n X X n q n5 n3. (3.6) : qq q q qq qq nn
187 X F X F X F 3 3 S X d X S34 X4 d4 X3 X F i i S56 X6 d6 X X F n n S X d X X F n n N S... S X d X... S34 S X d X N... q Sn n X n dn X n... S s3s N S s ss q q q q q n5 n4 n4 n4 n5 n3 n n n n3. (3.63) : K U X F. (3.64) V N S K U V F S : K nn qq qq... qq qq n n n n n n n n n n 87
188 s s U s s V n i n X X X X n i n F F F F s q s N N N N s s q q s S S S S (3.65) n n K. : S F U V K N X V U K U V K * * * * * * (3.66) * K * V * U. (3.66) : F U S V F K N X U U K U U K V V K K * * * * * * * *. (3.67). : F K UU F K S V K K UU K K U F U N F K UU F K S V K UU K K X v v * * * * * * * * * * * * * * ] [ ] [ (3.68) V V K K K v * *.
189 -.. s s n s. - 3s s s. - s N. n n n : q q q : N N N q... s. (3.69) i i i X i X X X i... n. (3.7) 89
190 . (3.7) (3.55) : q q X X q... s. (3.7) Nq F F b i R R R.. R R R.. R R R g i qq N : 3 3 F F F qq qq qq n n n n n n 3 F F F F F F b F F b b F n q q n n F N q F F F F b n 3 F bq b n N N bq b n bq n 34 N q N n n q n n R qq - F i i q N N N qq N n n. qq n n3 q q : F b i bi X i i i (3.7) m X. (3.73) - : i 9
191 Z R R R.. R R R... R R R 3 3 n n R R R q q qq q q n n n n f f n n n Z n. 3 3 R R R R R R R R R n n 3 3 q q qq q q n n n n R f R R R R n n R qq q q n n n n qq f n (3.74) R qq : R X R3 3( X X3 ) R3 3( X X3 ).. Rq q qq ( Xq Xq ) R qq qq( Xq Xq ) Rqq qq( Xq Xq )... Rn n n n( Xn Xn ) Rn n n n( XnXn ) Rn n n nxn. (3.75) (3.7) (3.69) - : 9
192 R R f F Fb N N R3 R3 F Fb N N R3 R3 F3 Fb 3 N34 N34.. Rq q Rq qfq Fbq qnqq Nq q R qqr qqfq Fb q q N q q N q q Rqq Rq q Fq Fbq qnqq Nq q.. Rn n Rn n Fn Fb n n Nn n3 Nn n3 Rn n Rn n Fn Fb nn Nn nnn n Rnn Rnn fnfn Fbn nnnn Nnn. (3.76) (3.56) (3.76). - : R f Fb N R3 Fb N R3 Fb 3 N34.. Rq q Fbq qnqq R qq Fbq q N qq Rq q Fb q q Nq q.. Rn n Fb nn Nn n3 Rn n Fb n n Nn n Rnn fnfbn nnnn. (3.77) : F m X m X i... n. (3.78) b i bi X i bi X i i i i i i 9
193 (3.76) - (3.77) (3.78) X i N q c : X fbx mx N 3( X X3 ) bx mx N 3( X X3 ) b3x 3 m3x 3 N.. X () X () qq ( XqXq ) bq X qmqx qnq... qq( Xq Xq ) bqx q mqx q Nq Xi () X i () qq( Xq Xq ) bq X qmq X qnq..... Xn () X n (). n n( Xn Xn ) bnx nmnx nns n n( XnXn ) bn X nmnx nns n nxn fn bx n n mx n n Ns (3.79) - - N : q qq N N q q - N q q N q N qq q N q... s. (3.8) (3.79) Nq - (3.7). (3.79). (3.79) : 93
194 94. ) ( ) ( ) (.. ) ( ) ( ) (.. ) ( ) ( ) ( n s n n n n n s n n n n n n n n s n n n n n n n n q q q q q q q q q q q q q q q q q q q q q q q q q q q f N X p b p m N X X p b p m N X X p b p m N X X p b p m N X X p b p m N X X p b p m N X X p b p m N X X p b p m f N X p b p m (3.8) i X q N Z f Z f n n n p i X q N f n f Z Z : ) ( ) ( dt e t X p X pt. (3.8) -. (3.7) : q q X X. (3.83) (3.83) (3.8) 3s 3s :
195 ) ( ) ( ) (... ) ( n s q n q q n q q q q q q f f N N N X X X X p K p K p K p K. (3.84) (3.84) - p :. ) (... ) ( ) (... ) ( n n n n n q q q q q q q q q q p b p m p K p b p m p K p b p m p K p b p m p K (3.85) (3.84) : ) ( Z N X V U p K f. (3.86) n n p K ) ( s s U s s V n X s N n f : ) ( ) ( ) ( ) ( ) ( p K p K p K p K p K n q q q q q q n n (3.87)
196 s s U s s V n i n X X X X ˆ s q s N N N N n n n f. n n p K ) (. - : ) ( ) ( ) ( * * * * * * Z U V p K N X V U p K U V p K f (3.88) * V * U * ( p) K -. (3.88) : Z U Z p K N X U U p K U U p K V V p K p K f f * * * * * * * ) ( ) ( ) ( ) ( ) (. (3.89) : Z UU p K p K p K UU p K p K p K U U N Z UU p K p K p K UU p K p K X f v f v * * * * * * * * * * * * ) ( ) ( )] ( ) ( ) ( )[ ( ) ( ) ( )] ( ) ( ) ( [ (3.9) V V p K p K p K v * * ) ( ) ( ) (.
197 - : N W ( p) Z (3.9) : W ( p)... W( p) W ( ) q p... W ( p) s W( p) U U K[ K K UU K] K K UU. (3.9) * * * * * * * v f N Z W q (p) Z - Z N q. - W q ( p ). W ( p) A ) ( q) ( W q ( p ) j. : A( q) ( ) Wq ( j) q... s (3.93) j ) : W q ( q( ) Dq ( ) j Wq ( j) (3.94) E ( ) F ( ) j ) D ) E ) F ). q ( q ( q ( q ( q - : q 97
198 A ( q) q( ) Dq ( ) ( ). (3.95) E ( ) F ( ) q q : q q q N N N q... s. (3.96) : q. q... s q A A ( q) ( ) q N q... s (3.97) (3.97) q. 98
199 q A A ( q) ( ) N q... s. (3.98) - : - (3.98) - : N q A min. (3.99) q A( q) ( ) s -. : A KP q Nq q... s (3.) A ( ) ( q ). - KP q A : KP q KP q A min A. (3.)
200 s A sin t H H i i... m i d i b i F i s. 3.3 q... 6 qq l qq 3.4
201 -. H - H H. -. H H : 6 H 7 H H.
202 . 3.. H : H- 7 ( 3.3). - H A ) 5. - ( q) (
203 : A ( ) ( ) A( 5 ) ( ) 5; A ) ( ) ( A( 4 ) ( ) 4; 3 A ( 3 ) ( )
204 : KP KP 5 KP A A 5; A KP4 A 4; 3 A ) 3 ( 3) (
205 : 5 : KP Nq A min. (3.) q...5 A( q) ( )
206 b i q q m i : mi 3mi. (3.3) bi b i q q q - 3 b i q 6
207 q q m. i i i 3 -. i : 3 i 5i i
208 : 3 i 5i i : 3 i 5i i 3 8
209 m ) - ( m : 3 4 m 8 3 m 6 9
210
211 (- ) ; - ; - ; - ; (- ) -
212 /.. //. -.. : /..... : /... : :
213 4.. - /.... : /.. //... : /. -.. : :
214 ; - ;. -. 4
215 ; : ; : 5
216 J - z XYZ -. - F F F n : d T T Qj ( j 6). (4.) dt q j qj q j x y z x'y'z' Oxyz. Q x Q y Q z - x y z Oxyz: 6
217 Q Q Q x y z F F F x y z y z y z ; (4.) Q Q Q - z r: Q M ( F ) M M Q M ( F ) M M z z z Q M ( F ) M M. r r r (4.3) Oxyz: ( ). T Mv J (4.4) - []: SinSin os ossin Sin os. (4.5) z : ij os os Sin Sin os os Sin Sin os os Sin Sin Sin os os Sin os Sin Sin os os os os Sin Sin Sin os Sin os. (4.6) 7
218 a dv dt v -. : = (4.7) = + r + ( r ) ; d Oxyz; dt yz; r dv O ; xi O yojz ; i j dt Oxyz. - : M a dm V dm- ; = + + ( ) ; V. V V V a dm Ma ( ) dm J e ( ( ) dm( J ) 8
219 : = J (J ). (4.8) -. : = Ma (4.9) a = v. : M a dm a dm. : a a ( ) V ( ) dm ( J JE) : = [(J JE) ]. (4.) : J = (J +J +J 3 )/; J J J 3 ; - (). (4.) (4.) (4.3) (4.) x y z yz (4.) (4.) -. (4.8) (4.) 9
220 ( ) max max [].. Oxyz Oz. D : = + r + ( r) O; r D;. D : n. n Oxyz : a a a x y nz a a a Ox Oy Oz y z x z x y y x y y z x y x x y y x z x y z
221 x y z D.... : a a On O O. O. n < : a x y Oz y x z x y z O: d / y x z x y z. < : a Ox y z x z x y y a x y Oy z x Oxy O. y x y
222 M M (4.8): B Sin B os z B B B 3 4 B os B Sin J J rz B Sin 3 B os 3 B J B os 4 B Sin Jll Jrr x y J rlx J J. rl y J x lz x y ll rz rr x x 4 J lz y y y (4.) J rz J lz... rlz ' ( r z ' Oz). (4.)..
223 «-» : ; ; -.. : ( - ). - 3
224 (- ) -. - () () -. : ) ( ) : :. :
225 5. E A ; -... z Oz Oxy; Ar l z Ar (. 5.). - : z z z. z z. - A ( 3) ; x y 5
226 x Ar. : P N F A ( 3) :. - (4.) (4.3): Mx My J os J J z J P 3 rp P z x P y N os r F rp F l F x y Sin l Sin J zl z F r x y z z r F M z l Sin l F z x y l F M r F M z N r l Sin M M r N M r M r M r (5.) J J J Sin os J J J J M ; rp z z ij Oxyz - 6
227 (4.6); r l A Ar l z ; F r F l. P x P y P z - M - (4.7) (4.). ( x y ) (5.) : : N N N l r P z M M Ar Al z M M A r A l Fr Pr r Fl Pl l r Fl lfr M z M z. (5.) (5.3) : = + = F f N 3 ( f N ) : F r F l l QA Q(r Q l Q ). Q F l r rq F (5.4) QA (5.4) Q A. (5.3) Q. (5.4) 7
228 . - - Q M Q z Pl r Q Pr l Q F QA M Pl Pr F AA F AA i Ai z r i l i j i j i F F i j i A i. (ij):. (5.5) (5.6) Aiz ( N ) ja + Na j Sign ri Pl + lipr + M Ai z M Ai z J = f + (ij):. (5.7) 3 i i Naj b j 3 j a a a ; a J l; a J r; i i i i i 3 i Aiz i 3i Aiz i a J l fj a Sign i j Aiz j Ai rz a J r fj a Sign b i 3 j Aiz j Ai lz i i Aiz Ailz l Air Ai r i 3 P z z b J J Pz M M b J J Aiz Airz Pz J rp lp M M ; Ai rz i l i r Ai z Ai z M M r Ai l Ai l J rp l P M M ; Ailz i l i r Ai z Ai z (5.8) J Aiz J Ai lz - rlz i. - i 8
229 F ir ( F ) + F < - il i (5.6) -. : F f v i - N i i v v / v i. - i i i : = = (5.): : M x M y J z P x P y fn f N fn v v r v l v M ; x y l x y N i : N N N P r x y z r f zvr M l M l M l l f z v M M M. z z l z r r r (5.9) (5.) - i v i = F i - F i. : x = y = = (5.5). (5.) (5.)
230 -. - ; A x y i. ( ) 3 i i i F (t) M () t. z A i F i. - A i. Qxy ( ) Oxy -. - : M t x y F QA M y y F x x F (5.) Qz i i z x y 3
231 QA ( xx ) ( y y ). i i i Q ( x y ) MQZ ( txy ) t 6. M ( txy ) A i - : y i i i x i i i QZ F F x x / QA F F y y / QA. (5.) : F F QA z / QA i i i i z ;. M Qz x M Qz x y M y Qz x y M. M ( txy ) - QZ M ( txy ) Qz QZ 3
232 : g x y F QA ( ) i i. [] - gxy ( ) F F = = F 3. F i. - gxy ( ) M ( txy ) F x = F y = (5.) F F F. 3 gxy ( ) - : F F F A A A3 : ) F F F ; 3 QZ 3 ) - Q (x y ) (- g( x y). ). 5.. Q g(xy) : i A A A ; 3 i - F F F (. 5.). 3 3
233 F F 3 R 3 3 ; 3 R - - F F. 3 3 Q g( x y. ) - Q - g( x y ): x ' x a a y y Sin / O O Sin Sin y ' y a a x x Sin / O O Sin Sin m R Sin Sin Sin F tg S 3 i i 4 F x x3 x y3 y ctg y y ctg3/ ; y3 y y3 y x3 x ctg x x ctg3/ ; OO x x y y ; (5.3) R=a i /Sin i 3 ; SF = Fi Fj Sin / a i =A j A (ij). Q : < ( i = 3) i i F i < F + F (ij). Q - A i : - i : F i j < F + F (ij) j < min g( x y) g( x y ). i i i i Q - 3 (. 5.); 33
234 (5.) = = y x F F. Q : 3 + > F F F ( ) ( ) ( ) ( ). g x y F a F a F F a F a F a F a a Fa Fa gx y 3 3 ( ) ( ). g x y g x y 3 3 min ( ) ( ) g x y g x y <. l os Sin g(xy) l 3 :. os F F F F F os F F F F F a a y y y y x x x x F F F F F a y y F a y y F a x x F a x x F F Sin a y y F a y y F os a x x F a x x F F Sin y y os x x a F Sin y y os x x a F F Sin QA y y os QA x x F Sin y QA os x QA F Sin y g os x g l g i i i i i A Q i i i A Q A Q A Q lim lim lim lim ( ) g x y... F (t).
235 M ( t x y) g( x y) M () t Qz z ; M () t - gxy ( ) Q (5.3) M z() t.. F (t). - Ox F F = F >. y = x z F = Fi. - ' r (r i i ): M ' z ' i i r F r / F. (5.4) ' =. M Qz txy gxy y yf x Fi QAi Fi yi y F i x x y y i i y yf / F. i F / F - F M ( txy ) F =F y=. F. 3. : M z F. - Oxy F y = M z F x >. (5.) : i i i QZ F ( x x ) / QA F F ( y y ) / QA. x i i i Q( x y ) : 35
236 : g g min x i i x max x. y maxm F y y ty M F y y. ' z i x i ' ' g( x y) g ( y) lim g( x y) g ( y) y x; M ( ) ( ) ( ). QZ t x y g y g t y g (ty) M = M z F =F =F x. y = y M / F < y ; g( y) g( t y ) (. 5.3): y y M ' F F y y m M F y y M ' F F ' - ' m ' ' - : ) M m ' Ft (5.5) F M ( txy ) QZ ; M Qz (txy) 36
237 ) F F F (5.6) m F M ' M ( txy ) QZ y m y y A i ); ) t ' F M F m (5.7) M ( txy ) - Qz ( - ). (5.5) (5.7) i (ij): F i F j +F +F Q MQz ( txy ) A i - - ). M ( txy ) A i (ij) Qz x i xj xi x y i yj yi y i i j x a a j a a j F F F F F F (5.8) M' z ( y' yi) F x. 37
238 F M z : Oxy Oxy : os F x F x x os y Sin i y x Sin y os i F x' Sign M i i F Sign M ' z i i ' z Sin F y F Sign M (5.8) - Q A i. x i y i F x. (5.8) i - - M. ' z ' z
239 f i. (5.9) ( Ox ): M x N v MgSin My f i i ix f inv i iy J N rv lv z f i i i il i ir (5.9) g. N i A i (5.): N Mgos i N r fzv i i i ir N l fzv i i i il. (5.). [3] : tg f - ; f tg f ) ; tg > f. - : z i i i il i ir. M fn rv lv (5.) z ' (5.) Mz M 39
240 =.. f i =f (i = 3) (5.9) : x = x + v t + at y = y = (5.3) (x c y c v a ) -. cz = -.. [] f = f( i ) = arctg( y / x ) i i Ox (5.3) i : f i ( i ) = f /[+eos( i i )] (5.4) i i f ; < e < ; i v Ar ; i i
241 i f i ( i )=A i B i - (e<<). - M z = (y i y )F ix = f i N i (y i y )= ef N i d i Sin( + i )os( + i ) d i =AA i ; i i Ar. - : F iy = f i f [ eos( i i )] i =. : = = 3 = ( i z = ). M z M z ef N d Sin ef N d os os (5.) : / < 3 < / = ± / - ; / < 3 < 3/ : 3 = 3 4 = = = ± / 3 4
242 ; M z. ( - ) : f = f = f f 3 = f +. - : M z yi y Fix fi N iyi y y3 y N 3 N 3d 3Sin 3 ; M z N3d3os 3. - =
243 . - - (4.8) D (. 5.6) - - = Sint t. Ox
244 G N F t z m zy yz m yy z n m zy y z. ; ; x yx y Oxyz ; x y z Oxyz. : mx mgsin F x my mgossin mz my mz F mz mgosos N my mz my. y (5.5) : v F f Nv v v/ v ; f -. - gos L = : t = = t ; T T = 44
245 ~ x y x ; ~ y ; ~ z L L z L ; ; ~ x y z x ; ~ y ; ~ z L L L ; ~ x ~ y ; ; ~ z x y z ; L L L ~ N ~ F N F. mg os mg os (5.5) ( - ): x" tgf x y Sin +z "+y ' z' '+F " y z os y "+z ' y' '+N ". N = F = F =. z = z' = z" = (5.6) : x" tgf x y Sin y F " ' y N os y" y' '. x' y' F fn F. y fn x' y' x' y' : (5.6): ' ' y (5.6) (5.7) x y F F F fn; (5.8) 45
246 F F x y tg Sin y N os y.. = ; Sin os y. N > (5.7) : (5.9) Y y <. (5.3) = F > f N. (5.9) (5.3): ( ) y g f yg Sint ( ) : h ( a) Y ( ) ( ) Y / g a/ f a / f. h () a a - : ) D ; ) a a ; ) a a 46
247 D = (a Y) 4 ( Y ) h (); a ( Y a D)/Y h (a) (a a ). ) 3 a Y 3 a ; (5.3) ) a a ) Y a a ; (5.3) a Y. (5.33) a F < f N. f y g y g : h ( a) Y ( ) ( ) a a Y < a. (5.34) (5.3) (5.34) Y (. 5.7): - ; - ; - ; V. - 47
248 Y : < - ; : (y<) ( y >) ( y =).. 48
249 49 -. : Y << Y << Y = t. (5.7) Y :. Y SignY Sin a f a Y (5.35) : a Sin Y=. - (5.35) : ; 3 = + 4 = +. (5.35) : ; f a os Sin Sin Y Y Y ; f a os Sin Sin Y Y f a os os Y ; f a os Sin Sin Y Y Y ; f a os Sin Sin Y Y f a os os Y : a Sin = ; (5.36) Y=: a (os os ) + =. (5.37)
250 : a Sin < (5.38). = + (5.35) : os a (5.34) a / 4. a / 4. - (5.34) (5.35). : Y a f Y asin. (5.39) Y Y Y. : my my mgysin n mgyos c mgyos - (Y<<). - (5.9) : Y a f a f a Sin SignY YSin SignY a a f Yos f os Y. (5.4) : Y- Y; Y 5
251 (5.4) : Y Y ; 4 Sin Sin (5.4) a (5.36) (5.37) (5.36) (5.37) : R 4 Y Y f - ; R f a 4 Y a f Y f ( ) (R f R ) 5
252 R - ( ): <5 >5. -. R f T Y : Y. (5.4) Y B Y A e B ; -. > Y - ; < = - ; < Y=B (5.7) = Y ; Y : Y Sin Sin f os Sin f Y Sin SignY SignY f os Y Yos 5
253 (5.7) (5.8). Mathad 3 rfixed f = = 5 = Y expfit A = 86 B = -4 = : ) 3 4 ; ) ; ) D Y; ). 53
254 . 5.. f = = : - - < < ; < 5 - > i (i =4) A : ; - 54
255 V x f = 55
256 : ) - - ; ) = (t) (. 5.). - i (r i l i ) (i = 3) - - : x i x i x x d os i d Sin i i ; yi y disin i ; ; y y d os i i i i (5.43) d i i i i AA r l arctg l / r ). - i ( i i : P; N i F i A i. - (4.7) (4.8): 56
257 M M M x x y z 3 J J Sin os ; 3 J os J J y J 3 My z Sin Sin os My (5.44) (4.9) (4.): M y z 3 M J J Sin x My J3 J os J M z. (5.9) (5.) : : M x M y J Mg Sin Mgos Sin f N i f N v f N v r v l v J J Sin os i il i i ix N i : N N r J N i i i l ir 3 i iy Sin J os J os J Sin. My Mg os os My My i i 3 3 (5.45) (5.46) : z c = J = J +J 3. : N i M i gos os y y J os Sin J S S l S i Sin os ri S (5.47) S i AA j A ; r i l i A j A Ar Al ; i j - ( 3); S 3. 57
258 - N i. - - gos os y y MS i. (5.44) rj os lj Sin. * * t t i i : max max L max( i MS y i i r J l i J 3 ) gosos L. (5.48) (5.45) - (5.46). : M z J J Sin os. 3 J < J 3 (5.44) - - = / = 3/ (4.): M z M z ( J 3 J ). 58
259 (. 5.) : ) ; ) i (i = ); ) i (i = ) = Sint t. Ox 59
260 . - x y r Ox. - G i (i=) - myz m yy t n i i i i m yz. c i ; ; x y x y x y x y l Sin l os l Sin l os x y x y x y x y i l os l Sin l os l Sin i Oxyz (i=); M M =l - ; x y z Oxyz. A i (i=) - =l. A i N i F i (i=). - ( z z z ): mx my mgsin F mgosos N N my 4my N l N l my l my l my l my l ml F r l F r x F x mgos Sin my l my lsin my lsin. F y F y (5.49) 6
261 : vi F fnv i i i f ; v v / v i i i i; v x v x x x l os l os i ; F i r v y v y y y F os F Sin i x i y l Sin l Sin r. - T = gos L = : t = = t ; T x y xi yi ~ l x~ = ; y~ = ; x~ i = ; y~ i = ; l = ; L L L L L ; x v y v y~ ' x y x~ ' = ; = ; v~ x = ; v~ y = ; L L L L x y = ; y~ " x~ " = ; L L N F N ~ F ~ i = i =. mgos mgos (5.49) ( - ): 6
262 x" y" tg F -Sin y ' - os - y "- y' - l" os l' ' Sin - N N " -' x F x F Sin os l y F ' N F r y N - F r. (5.5) N i (N i ; i=) : N [ ( ) ( )] = os + y" +y' ' - l " os - ' ' Sin N = [ ( os + y " y' ' ) l( " os - ' ' Sin) ] + + (5.5) v v ( A i ; i=) - ( - (5.5)): F ix f N i v x" y" = tg + F = -Sin + y ' + F " = -' i x f N x + F x y + F y Sin os + ( F r - Fr ); l i v v ix ix v iy F iy f N i v i y f N i v v ix iy v iy. (5.5) v = v F < f N ( A ) : ( ) [( ) ] y + l os ' Sin + F r Fr " = - + l + (5.53) F r = f NSign' Fr = tg os - Sin Sin ; - : 6
263 F F x y = x" = y" - tg - F x = - l + Sin - y ' ( " os - ' Sin) - tg - Fr ( ) - F y = - l " Sin + ' - y ' - F r Sin. os os + Sin - (5.54) v v = F < f N (- A ) : ( ) [( y - l os ) ' Sin F F ] + r " r = l + + (5.55) F r = f N Sign; - : F F x y = x" = y" - tg - F x = - l + Sin - y ' ( " os -' Sin) - tg - Fr ( ) - F y = - l " Sin + ' - y ' - F r Sin. v v F fnf fn r r f N F f N F F r r l os os + Sin - (5.56) F l = tg Sin Sin os - - (5.56) : F F r r = - y = - [( ) ] - l os ' Sin + F [( y l os ) ' Sin F ]. + + r r (5.57) (5.56); -. F r < f N 63
264 (5.55) [( y - l os) ' Sin + F ]. F = - f N Sign (5.58) r F r < f N - (5.53) - F [( ) = - f N Sign y + l os ' Sin + F ]. (5.59) r (5.56) (5.5) : r r F F l ix = f N - Fr SignF l F l = f N - Fr SignF l = F os - F Sin F = F Sin + F os ( i = ). ir il iy ir il (5.6). A (v v = ) F < f N (5.5) : v x v x = os Sign ' v = Sin Sign ' = - F F x x + F y v y y = - F F x y + F y. (5.6) A (v = v ) - F < f N - (5.5) : v x v x = - F F x x + F y v y = - + F = - os Sign ' v = - Sin Sign '. y F F x y y (5.6) 64
265 (5.5) (5.6) Mathad : - ( = /) Sin os (5.5)) - ( = ).. - ( = / =85). 5.3 ). ( /): 4 45 = - ; ; 65
266 45 ; =/ - ; 3 75 =. ( 4) - i = = 5 (. 5.3 ). ) ) : a) l = ; f = ; a = 85; = ; ) l = ; f = ; a = 5; = 66
267 ) ) ) : ) l = ; f = ; a = 85; = ; = 4; ) l = ; f = ; a = 85; = ; = 4; ) l = ; f = ; a = 85; = ; =
268 (4 45) - (. 5.4 = 4) -. = 4 - =. 5.4 ) ; = - = = 463 = ). = = D (. 5.5). - - A Sin t. : t. - ( D). 68
269 B. - J. - x y - ; h l D l B : ) ; ) : - D B B. : G N D N B F D F B D B. - D N D F D. : F D f N D ; F B f N B. f. : Mx My J F N D Dx F N os Mg Sin F F h N l N l Dx B Bx Bx D D B B (5.63) = MA Sint ;. 69
270 N D = N B = -. - N D N B (). - x <. : N D > N B > F Dx +F Bx = f (N D + N B ) x > = y =. (5.64) (5.63) : Mx f(n N ) os D B N N Mg Sin D B f(n N )hnl Nl. D B D D B B (5.65) (5.65) N D N B x (t) ( t ) : x M( lb fh )( g A Sin Sint ) N D l l D M( ld fh )( g A Sin Sint ) N B l l D x ( t ) A( os fsin )(( t t )ost Sint Sint ) v (t t ) x B B x ( t ) A ( os fsin )( ost ost ) v. (5.66) v x - t. N D N B - g < A Sin g = arcsin A Sin t g > A Sin (5.67). 7
271 - : x N N F F f N N. (5.68) D B Dx Bx D B (5.63) : F Dx FBx os N N Mg Sin D B F F hnl Nl. Dx Bx D D B B : (5.69) M( gl B ( lbsin hos )A Sint ) N D l l M( gl D ( ldsin hos )A Sint ) N B. l l (5.68) : os < f (Mg Sin) D D A ossint f g A SinSint. B B (5.7) (5.7) : = t z=g/(a ) -. (5.7) ossin f z Sin Sin t. (5.7) - fz Sin < fsin + os fz > f Sin + os 7
272 z > Sin + os (5.73) f (). : N D = (F D =); N B > F B <f N B F B h> N B l B. (5.74) lb FB < < f h N l B f h <. (5.75) : J B B y os x x Sin Mgx x ; (5.76) B F Mx os N My Mg Sin. Bx B (5.77) (5.76) (5.77) : x B x x l cos hsin y l sin hcos (5.78) y x x y x x y B B. (5.79) F B < f N B : Mx My fn N J N B B B ( fh l ). os Mg Sin (5.8) B B 7
273 (5.78) (5.79) y y x (5.79) x B ). (5.8) x (t) (t) N B ). : J J J ~ B = J ~ B = ; MA MA ~ l = l / A h ~ h / A ; B B = d d ' = = / d d ; N ~ / ; " = = N F F ~ MA MA B B B = B =. (5.76) (5.77) ( - ): J B l Sin hos zl os hsin Sin (5.8) B B F Bx l Sin hos l os hsin Sin os (5.8) B B N B l os hsin l Sin hos. Sin Sin z (5.83) B (5.74) (5.75) - ( = = =): Sin Sin z N (5.84) Sin Sin z F f Sin os f (5.85) B B B B N B Sin Sin z F h N l h Sin os l (5.86) B B B < f (5.84) l B. (5.87) h 73
274 z Sin < Sin z > Sin. (5.88) (5.85) f z Sin f Sin os z > Sin + os. (5.89) f (5.86) z Sin > Sin + os z < Sin + os. (5.9) (5.89). B D. - : z Sin = os + Sin (5.8) (5.83). ( < < ) = = (5.8) (5.83). : 74
275 F N ( ) J " = hsin os + Sin - h z B Bx B = - Sin os + " h = - Sin Sin + z + " h (5.9). N B > - : JBSin Sin J Bz h Sin os Sin z. ( ) z Sin os (5.9) J h J J h h. B F Bx < f N B : J B Sin os h f J B Sin Sin J z h Sin os Sin B z Sin os Sin z P( ) Sin < zq( ) (5.93) P Q os f Sin f Sin os f f. P() Q() - ( f ) Q() > P() ( ) ( f ) P(). < < < (5.93) - - ( ) ( )( f ) P( ) z> Q( ). (5.94) 75
276 5.5 z h f. : ; - - ; ; 3 B D- ; (5.8). z t arctg ; N B F B f N B - (5.94). 3 - (5.9) 4 - (5.94) 3 4 (5.89). - (5.94) (5.94) 3 4. (5.9). Mathcad : ) ) ) h ; 76
277 ) ) ) ; ) ) ) ; ) ) ) f.. - : z ) - ) A g / z g = 98 ( ); h h A ); ha (); J h MA ( ) ; t (c). l B 77
278
279 R. - - P (. 5.6).. - = ASin t A t. - M J. x y Oxy. 79
280 - N F (- ). : F f N f. - N. : = M A Sin t. (5.95) ( ):. G : M x M y = F J = - F x + os = N - Mg + Sin x y - NlSin. (5.96) -. : x = x - lsin y = R - los (5.97) D x D y D = R D l = D. P v P = x = R y =. (5.98) D D 8
281 (5.97) (5.98) : x ( R - los) + lsin = (5.99) (5.96) : F x M - A Sintos R - l os lsin. (5.) = l Sin + los (5.) y : N M l Sin l os - A Sin t Sin g. (5.) (5.) (5.) (5.96) : A Sin t Ros - l os - g lsin - Rl Sin (5.3) J R - l os l Sin M Ros - l os - Mg lsin - M Rl Sin (5.4) J P J J M R l os Ml Sin P ( ). : = t ; F ~ F = ; MA N N ~ = MA ; z=g/(a ) ; J J ~ P P = MA ; h ~ = h / A ; 8
282 R ~ = R / A ; ~ l l / A ; d d ' = = / d d x" = / ; " = = ; x d x = A d. (5.) (5.) (5.3) ( ): F ( R - los) + ' l Sin = - Sin os + " (5.5) N = " l Sin + ' los - Sin Sin + z (5.6) " = { Sin [ Ros - los( + )] - lsin [ z + ' R] }. (5.7) J P. - - : X = x X = x' X = X 3 = '. (5.8) : dx d dx d dx d dx3 d X 3 M R l osx X l SinX X M 3 M Sin R os l os X lsinx z X 3 R. J P (5.9) 8
283 (5.9) F x < f N (5.) F x N (5.5) (5.6): F Sin os M R l osx X l SinX (5.) x 3 N M z. l SinX X 3 l osx Sin Sin (5.). - : F f N Signv (5.3) x ( ) ( ) vpx = x' - ' R - l os = X Px - X R - l osx (5.4) Ox (- Sign v x = Sign F x F x (5.)). (5.96) (5.) (5.3) - - : 3 x" f N Sign v Sin os (5.5) Px N " = ( f Sign v ( R - l os ) Px - l Sin). (5.6) J.. (5.6) (5.6) : ( ' l os - Sin Sin + z) ( ) J N =. (5.7) J - f Signv l R - l os Sin + l Sin Px 83
284 (5.5) (5.6). (5.8) : dx X d dx f N SignvPx Sin os d dx X 3 d dx 3 N f Sign v Px R l osx l SinX d J J X 3 l osx Sin Sin z N J f Signv l R l osx SinX l Sin X Px (5.8) (5.9) v x (5.4). - (5.8) v x = - (5.) (5.). - (5.). (5.9) - f (- ) : J f <. l R -. (5.3) - 84
285 = ( = ): " = - J z l R - l + + ( R - l) J + ( R - l) : Sin. zl. (5.) J R l ~ = (5.) ~ = : J R l (5.) l z. : l = = R (5.) : J R. R z. (5.). (z) ~ =
286 V Am = max / : - Am - 86
287 z Mathcad 3. : R ); V ; - ); Am (). z z - ( - )
288 : R V- (. 5.8 ) Am - z (. 5.9 ); ( /6) V (. 5.8 ) - Am z = 5 (. 5.9 ); - V z = 5 - z = 5 (. 5.8 ) Am - z (. 5.9 ); V z = 5 z = ) Am < z < 5 z > 5 Am(z) (. 5.9 ); f V - z = 5 z = (. 5.8 ) Am 7 < z < f. Am(z) Am. 88
289 : : : -. : :
290
291 - () ( ). (. 6.) L L : - 9
292 L L ; - L L ;. -. : Sini l tg i i Sin i l i A A. l i A A : l Sin Sin l Sin Sin Sin Sin l l l (6.) Sin tg. l Sin µ i i (i = ). - - (6.). ; ()
293 L L. -.. A A - A (. 6.) x y ; A A z A const. - l i
294 F (t) M z (t). (- ) - A A Ar l Al A r (. 6.3). A i F i - F i : F l F r M : F F z l r F l F r (6.) l l F l l F. r r 94
295 F r l l M z F F r M z F l l A : r r ( l l ( l l c c ). ) (6.3) F ; F F F F (6.4) r r v A : l v F r r F r l r l. A : ; F ; F F F F (6.5) r r v A : l F r F v r. : F r F F F F F F F F ; r r r l (6.6) i - v i : F F i F ir ir i l Fi Fi v r SignF l i. (6.4) (6.6) F F F r r l ( F l ) 95
296 . 6.4). I II III. - F F F r r l (6.3). - A ; II A ; III ; I ; - 96
297 (. 6.5). - - f. A. - M J J J 3 (J = J ); - x y. - ( ): Sin( ) AA. os : - G N i F i A i (i=); x y A i : 97
298 F v v ix iy fni Fiy fni (6.7) v v v v ix ix iy ix iy v v ix iy x y li l os l i os os l l Sin l os Sin i i (6.8) A i ; l i =AA i ; l =ASin. F i < fn i. (4.4) : J T Mv J 3 (6.9) v ; (4.5). (4.) (4.3): F i r Q Q Q Q Q Q x y z Mg Sin F F Mg os N y F F r N F y F os F i x F N rl F rl os l l F r l l l l N l l F F z i y x x Sin F il F i x l l Sin F os. i y (6.) (4.) z :
299 M x M y Mg Sin F F y F Mg os N J J os 3 J os J 3 J Sin 3 3 N y J F z z J x F N F l F l r os rl l F r l l Sin os l l N l l F F z 3 x r l l (6.) J Sin J os. J z z 3 h = A = A T / g. h - : = t ; x~ x / h y~ y / h z~ z / h os ; dx ~ x dy ~ y x~ y~ ; d h d h d x~ x d y~ y x~ y~ ; d h d h d d d d ; d d d d ; 99
300 M h j J / Mh j J / 3 3 ; ~ N ~ N / Mg N / M h F F / Mg F / M h ; i i i i i i ~ ~ l l / h l l / h Sin (i=). i i -. (6.) : x y Sin F F os N j 3 y F j os 3 y x j os j Sin 3 3 N F N j j F z z x F l F l j r os rl Sin F r l Sin Sin os l Sin N l Sin F F os. 3 r l l (6.) A : xa ya os. x Sin os (6.) : - j y Sin Sin Sin (6.3) A l - : j l l os Sin Sin j tg j A l l l j os j Sin ; N i os l j Sin os Sin Sin ja l tg (6.4) l l j i 3 3
301 (i j = ; ij); - (6.4) (6.6) : F F M F r l i z j j l l Al Sin Sin Sin j j j fn i Sin os Al 3 Sin os Sin os i. (6.5) A i - (6.7); - (6.): N N N i os l Sin f os j j i il 3 Sin os j Sin 3 (6.6) v il Al A i. (6.) -. - : x const y / (6.7) ; - Ox. tg f (6.7). - : F ix f N i F iy f Ni x c y c l l i l os. i 3
302 (6.) : x Sin f ( N N ) Sin f os x x const. : os (6.). - - : y a b y b 3 a b y b b 3 (6.8) a3 b3y b3 b33. f N fmn fms b b 3 b x x x jsin fr f( j S lmn j3n Sin ) b 3 b 3 x jsin jj3 x (6.9) f( j lrn j3mnsin ) b 33 aj bjsin j 3 jjx j j 3 b b x j 3 N N R N N N l ; N M l N ; N R S l N R N l ; M Sin M S N M N Sin ; Sin N Sin ; j l j os j Sin. 3 (6.8) - (6.9) - a ij b ij : 3
303 b b 3 b Sin b b 3 b b 3 3 b : 33. a a 3 b b 33 b b b 4 3 a a a a (6.) b 33 b b x 3 b Sin 3 a a 4 3 b b b b 4 33 x b 3 b 3. Sin - (6.) : a a a a a a a a a a a. (6.) (6.) : f N Sin tg fb 3 a4 B a x jj 3 x jj 3 f (jx B f j Sin BB) aa a 4 3 Al 3 4 x j j3 Sin f (jx BB f Sin BBB) aaa a aa x j j3 Sin B R N M N N l l N N B j3m SRN jm NRSSin os Al 3 N S S jr 3 N B j3 j N Sin j3m N j lrn j 3jN jos R N Sin B Sin os B j N M B j B N Sin os B S R R Sin M Sin Sin N l l Sin N l l Sin. (6.) (6.3) 33
304 (6.) (6.) 4 : v f j A l 4 Sin BB j B 3 (6.4) v f Sin j 3 BBB B B (6.5) v x. (6.) B B 3 > B 4 > B B 3 < B 4 > v x B B 3 > B 4 < v x (6.7) B B 3 B > B > - 4 > > (6.). (6.) B > B < - B B ) B 4 > : B3 N Sin os B / j l A. 34
305 v v v v - B < B 3 >. - : (B < B 3 > B 4 > x v ); (B > B 3 > B 4 > ); 3 (B < B 3 > B 4 < (6.)); 4 (B > B 3 > B 4 < x v ); 5 - (B < B 3 < B 4 < : ' ' x v x v v v ); 6 - (B > B 3 < B 4 < - (6.)). (6.8): I. B > B 4 > (. 6.5); II. B < B 4 > v x (. 6.5); III. B > B 3 > B 4 < v x ( ). x II III : IV. ) B < B 3 < ( ); ) B B 3 < B 4 < ). I II III IV j j 3 J J
306 m r.. J m r m J3 J 3 J. - r ( ) ctg. J 3 ctg ( J M A ). ctg A m r : j j 3 j j 3 tg tg j. B B 3 B 4 (6.3): MS RN ) B j j3 M R Sin os N S ;. ) B 3 ) B 4 B j 3 Sin os NM ; j S 4 Bos j3m NM Sos j3 NM SSin. M os ( M os j N Sin ) N N 3 B M S R S < - ; B 3 M S >. - M S R S -. (6.6) (6.3) : 36
307 N N N os ; M N l N l Sinos ( f tg ); N M S M NSin N Sin Sin osos ( f tg ) R R Sin M Sin. N S S M S RS Sin ; B B M S < B 3 > M S > B <. M S f = tg tg ; M S. B 3 B 4 (6.3) B 4 < B 3 =. j j ; 37
308 -. (. 6.6 ) x ' * v. 6.7: (6.) ' x ; ' y ' ' y. - v * j j 3 4. (. 6.6 ) - - ' x ; (. 6.6 ) ' x ; - 4 (. 6.6 ) x ' v * (. 6.6 ) 3 ' x ;
309 ) ) ) : ) f = = 47 l = 433 l = 73; ) f = 376 = 47 l = 4 l = 865; ) f = 445 = 47 l =.579 l =.97; ) 39
310 ) ) ( * ' ' * ): ) j =3 j 3 = v 54 x 3 (x v ) =4 =; ) j =3 * ' ' * j 3 = v 54 x 5 ( x v ) =4 = ' ): j = j 3 = x 5 =4 = 3
311 ' ): j =7 j 3 = x 5 =4 = ) ) * ' ' * ): ) j =6 j 3 =4 v 7 x (x v ) =4 * ' ' * =; ) j =6 j 3 =4 v 7 x 4 (x v ) =4 = 3
312 ' ): j =8 j 3 =4 x 4 =4 = (6.) - ' ' x x const. (6.6) (6.) - (6.). x ' * v ' x - v * ; - x ' -. tg f (6.6) -. 3
313 -. tg f I IV. (6.) () A A. (. 6.). -. l i =A i A (i=); - J. x y P - N i 33
314 F i A i : Mx My Mz J J J 3 z F F N F J J zr i r z x y F F N J J P P P Sin l os os F Sin os N Sin Sin M i zz x y zr Sin li l Fil ASin Ni li l Filz J J Sin J pp zr z x y A zl i l A F ir i A M z M r. (6.7) F ir F il r l z r z z; J J J 3 ; J J J J J J J z z r z z r r z l p p J os J J J J J J J J os J Sin 3 3 J Sin os Sin J Sin J os. 3 J / J Sin J os Sin os (6.8) F i Oxy : F F ix iy fn fn i i v v v ix v ix ix v iy v iy iy (6.9) 34
315 v v x y z ix iy i i x x c y c y c c ( l osos ) ( y y 3 ( l Sinos ) ( x x l Sin 3 l os A A A A A A i i 3 33 A A i i A i (i= ); A A ; 33 xyz (4.6). : os v v ( v v F i fn i x y ( i ). (6.7) : J Al x i y i c c ) ) ) (6.3) Sin ( J os M l ) P z P Sin M (6.3) J Al zr A 3 J Sin os J os os J Sin M( z 33 3 : r z A Al Sin ); A N N i N ( l i l 3 M ) P z ( J ll A l J J rr 3 zr J M pp P A os Mz z ) Sin M r. (6.3) - (6.4) (6.6) : 35
316 F r F l M M Sin Mz Sin P Mz Sin os P ' z M z Sin Sin os P Sin J zl A A 3 Sin J zr Sin. l r l A (6.33) (6.7) = 6.5. P MgSin P P Mgos ; M. x y z - = (6.3) : arcsin tgos Sin tg arcsin os Sin os A A tg tgos os Sin ; (6.34) os os tgos A A tg os Sin A A A. (6.35) ( os Sin ) tg - : Mgos ( os Sin ) tg os A A A J AlSin Mgos ztgsinos J Al (6.36) z Aos Sin Aos. 36
317 i - (6.7) : x x arcsin v t a t y y f os Aos f os arcsin f os A f os (6.37) x v y - x x x v y y y ;. os A. (6.38) f ( os Sin ) A A y (6.37). - (6.38). - : - j j 3 ( j =J /Ml l = l l ); - ~ A ~ ~ ~ ~ A l l A A l ( l l ) l l l l los Sin( ) os( ) los ; A A f. (6.7) = (6.7) - 37
318 M M - nep op (4.7) (4.). - (6.36).. nep op A A (. 6.) A A ; J -. - x y
319 . - P N i - F i A i.. : Mx My Mz 3 F F N J J zz F F N ( F J ( J J zr x y zr J J zr x y P P P F )z ( N F F ( F rr r z x y r r r ) N N N l F ( F ) Sin M l l A ) Sin M A F l )z z M r. (6.39) F ir F il r l z l z z; J J J 3 - ; J J J zr zz rr ( J J )Sin os J Sin J os J os J Sin J ( J J J3 ) /. F i Oxy : F F ix iy fn fn i i v v ix ix v v ix v iy v iy iy 39
320 3 ) x x ( Sin z y v ) y y ( os z x v i iy i ix A i ;. os z z os Sin Sin y y Sin Sin os x x A A i A i i A i - : x y x y i i v v ( v v ) F fn ( i ). (6.39) - : Q A A Q M Sin z M J (6.4) ) ( 3 A Q z Sin M J J - ; :. M P z N N P ) ) z z ( Sin M ( N N M P Sin F F P ) Sin z M ( F F P F F r l z A A z l A r r r A r r l l l (6.4) : ) ) ) ( ( ( ) ) ) ( ( ( r z l A A r z l A A M P P z z z Sin M N M P P z z z Sin M N
321 F F F r r l ( M ( z ASin ) Pl ASin Pr M ( M ( z ASin ) Pl ASin Pr M F l P. l A i (6.4) (6.6) : F fn ( i ) F P. i i l l Px MgSin P y Pz Mgos ; M. : u z A u arcsin arcsin (6.4) u A u za u ; u tg os. A u - : ) ) Mg Aos z A tg os. J p (6.43) A - - : J p z A tg. Mg Aos (6.44) A A i (6.4) (6.6) : 3
322 Fi f Ni ( i ) F l MgSinSin. A i (6.39) : x x v arcsin t a t f fos y os y z arcsin A A f os f os. y. - - (6.39) - Mnep M (4.7) (4.) (6.44).. op 3
323 (- ); (. 7.). : q m n 33
324 : q m n q m n. 3; 3; 33 zc zc zc (7.) : x x ( q m n )/ z y y ( q m n )/ z z q m n (n m q). x c y c z c. x y A. - : x x c = yy c = (7.) z z c = ij (i j=3) Oxyz (4.6). Oxy Oxyz z z = - q = m : x y z x y ( m ( m m n n ) ) Sin n / z / z os. (7.) 34
325 J J ;. - - (J J ) J J J 3. : - ; N F-. - (4.4) : T Mvc ( J J J3 ) v ; Sin Sin os os Sin Sin os. (r c ) - K : 35
326 n n dk Mr F K mc F. dt F (= n) m c (F ) -. Oxyz ( z ) ( r ) ( ): M x M y M z P F x P F P N z y x y J z J J 3 z J ( J J J zz rp rr rp J Sin pp J J J J Sin J zl J rp J J ( J J zr rr rp zl 3 rp )Sin J os pp M rp Sin ( J M z )Sin r M pp J rr J m n Fr z m n N z m Fr z 3 Sin )Sin Sin os Sin os F z l (7.3) J J J z rr zl J J os J 3 ( J a 3 33 J J pp rp ( J J Sin )Sin os )Sin os J zz J a J zr 3 J rp J J a Sin pp 3 J Sin J J 3 a 33 os r p r l z ; F r =F x os + F y Sin F l =F x Sin + F y os r l ; M M z M r. 36
327 (7.3) (7.) : N Mz m n Pz M 3 z m n M z 4 4 m Sin n os Sin os P. z (7.4) N. - : v v c + A = : x y m z m z a 3 z Sin a z os 3 y y x x. (7.5) x y (7.3) : F r m M z m n Sin M 3 z M 3 z m Fl Mz M z m n Sin os M 3 z os m Sin n os m n os P m n Sin M z Sin os P. r l (7.6) F r F f N. (7.7) l (7.4) (7.6) (7.3) - : 37
328 J J J 3 J J J 33 J J J M J J J 3 3 J J 3 3 J 3 r M m J J 3 z J 33 n z 3 33 m M J n z P Sin os P Sin os z 33 m z r P Sin r P z l (7.8) J J 3 M m z 4 Sin J J J 3 M m z m n Sin os ( m n ) J Jzz M Sin os z J J JrpSin 3 3 M J J m Sin n os J ( ) rr z 3 M z J J Sin rp 4 m n Sin os m J ( J J J ) Sin M Sin ( z m n ) J 3 pp rr 3 4 z J J 33 rp J rp Sin mn J J J J Sin M m n Sin os 3 ( rr pp 3) ( ) 4 z m n J J M ( z m n ) Sinos 4 3 zl 4 z 38
329 J Jrp os 33 J ( J J J ) Sin Mm Sin 3 pp rr 3 J 3 33 J rp 3 J J zl M ( m n ) Sinos 3 mn J33 M ( m n ) Sin os. 4 z P MgSin P P Mg os M ; x y z P P os P Sin P P os P Sin. r x y l x y (7.8) - - ( ): os m n z Sin os z tg Sin. (7.) = / tg tg ( v ) tg + v tg = v=n/m. : tg ( ) 4 tg. (7.9) tg ( ) 4 tg :. (7.9). 7.): * v v* (7.) 39
330 = tg( /) *= tg(5 arctg f ) v*=( )/(+ ).. ( ) v = v* - ( = ); v > v* =n/m (7.9) - tg v. (7.) = / =. : = = / = -. - (7.8) : J J L J J J 3 L J J L (7.) 33
331 m L z L L 3 m n z m n 3 z gsin Sin MgSin Sin os Mg os m os tg. (7.) - J L J J (7.) J ' L3 J n J J J J / J J M os Sin 4 3 ( pp ) J z 3 L L J L / J MgSinSinos. J z (7.) : a 4 +a +a 4 = ' ' ' ' a J J J J a J L J L a L. L a a a a 4a - a 4 4 : a a 4aa4 / a. (7.3) : v v x y F f N F f N x y v v v v x y x J y n 33
332 v v ; - (7.5). (7.3) : x = x (t) y =y =const = = const = / = = const (7.3) -. : v ( v ) 4v f tg ; f y c (7.3)
333 U U-. : R ; R = R ( ); r t (t ) - A (. 8.) - : = r os t = r Sin t. (8.) J - Oxy 333
334 x y. : ( ) () : Mx My os t N Sin F G Sin t N os F J F R os Sin (8.) = r ; F ( - f ); N ; M - ( ); G=g - ; -. - : 334
335 v x = Sin y = os R x R os y R Sin (8.3) x R os R Sin y R Sin R os. (8.3) (8.) J MR Mg R Sin M R r os t v (8.4) : R N M g os rv Sin t ; (8.5) M ( ( F ( )) ) J g Sin r os t R. (8.6) J MR (8.3) (8.6) : x = y > x y R x y R; (8.7) J M R MgR M R r Sin t F J v (8.8) N Mr ost (8.9) v M J g rv Sint R. (8.) M R -. ( v ): x = Sin y = os (8.) v x os y Sin x Sin y os v R ; (8.) 335
336 : v Mv MgSin Mr os t f N sign( v R) (8.3) J f NRS ign( v R) ; (8.4) (5): v N M v gos r Sin t. (8.5) (8.) (8.5) : x v y x v R (8.6) Mv Mg Mr Sint f N S ign( v R ) (8.7) v J f NRsign( v R ) (8.8) N Mrv ost. (8.9). : ~ ~ ~ ~ d x t x x / r y y / r x x /( r ) v v v d ~ ~ d y d d y y /( r ) / / v d d d / r d d / r R R/ r j ( J MR )/ ( Mr ) j J /( Mr ) p N N / Mr F F / Mr z g / r / Mr. ( ) ( ) ( v ) ( v ) : / ; u( ) / / x y ; x ( ) x y arcsin x y y ; d / x y 336
337 ( - ): ( ( )) (8.) d R zsin os d jp R d N u( ) z os Sin( ) (8.) d j F ( z Sin os( t)) R; (8.) j j P F f N N. (8.3) (8.3) -. N - x os y z Sin. (8.4) d x 5( y y ) Oy; d N > N. P (8.) (8.3) F f N N> : ' v z Sin os( ) f N S ign( v R) (8.5) j f N RS ign( v R) (8.6) v N u( ) z os Sin( ). (8.7) 337
338 (8.5) (8.7) v R N (8.3) (8.4) N (8.) (8.) v R F f N N. (8.8) : ) = ' ' d= d = N=N > x Sin y os (8.9) - ; (8.) ' x ' R ' os y R' Sin F f N ; (8.3) ) = p N ; (8.4) d < ; ) = s (8.9) ' ' d= d s = N=N s > x Sin y os (8.3) s s s s 338
339 - ; (8.5) (8.6) v R Mathsoft Mathcad : z=5 R=95 =5 j =36 f=. =
340 ( ) : ) ; 34
341 ) - ; 3) U- R. r v - t t - : = r v ost = r v Sint. (8.3) Oxy : dv gf (8.33) dt = m m (- > ); g - g = 98 ; v ; F - ; a - ; a. 34
342 : F P( xy ) v (8.34) P(x y ) ; v v v os Sin x y v. - - F P x y os Sin v. py xysin (8.35) P xy (8.36) e y p( y) p( ( )) y ( R y h); h h ; p ; e(x y) ( e(x y) ). M(x y) MP v os Sin. p(y) -. e(x y)
343 e ( y y ) e( xy ) R x (8.37) R 4 {x = y }. ( y y) b( xy) x R 4 (x y) {x = y } ( b(xy) ). e(xy) : e xy e b xy. (8.37') ( ) ( ) ( ) P ( xy ) pb ( xy) e( xy) x y y (8.38) vx vy b( xy) R R. (x y ) 343
344 . x = y (x y ) ( y ) =. - (x y )= (x y ) + (x y ). (8.39) Oxy (. 8.3) - d x P( xy )v x rv os t dt d y g P( xy )v y rv Sin t. dt : (8.4) dx ~ d x~ t x~ x / rv y~ y / r x~ v x /( rv ) x~ x /( rv ) d d dy ~ d y~ y~ y /( r ) y~ y /( r ) P ~ ( x~ ~ v v y ) P( xy )/( rv ) d d x~ x y~ y vx os v y Sin x~ y~ x y x~ y~ x y h ~ h / r z g /( r ). v v (» - ): d x P( xy )v x os d d y z P( xy )v y Sin. d (8.4) x y. x y (8.4) 344
345 F x os z F y Sin. (8.4) - : F os y x F zsin. (8.43) F z zsin (8.44) - F F x y v os Sin ; F F F P(x y ). (8.45) - Oxy (. 8.3). ( ) p( ) P( ) P( ). (8.46) e Sin (8.45) P (8.46) ; F (8.44) = - z (: = 3; = 7; z = 39; = 3; = 5). 345
346 v os Sin ( /6; 7/6) F P P F. : ) F - ; )» ; 3) - ; 4)» ; 5)
347 : p( ) P P( ) e( ); P re ( ) e zesin z e Sin z < - z >. r e (). r e p e os e esin Sin os z e e e e (8.47) ; pp. (8.47) : pe os Sin ( ) / e. z = ± /. pe ( ) / e z ( e ) z p ( e ) z (8.48) e e 347
348 pe 3 4 arcos / e. z z pe e. (8.49) 3 4 e z r e 34 (- (8.48)) = ± / r e (): pe z ( MM M) = / e = / r e p ( ) z (8.5) e p r e ( ) z ; (8.5) e pe z ( MM M) - e (8.5) (8.5). (8.48) 34 r e () (8.49) (8.5) (8.5). (z p) - : ) (8.48) r e ( 34 ) > e p z z ( e e ) ; (8.5) 348
349 ) (8.48) p ( e ) z r e ( ) > e p ( e )( z) ; (8.53) ) (8.48) p ( e ) z r e e ( ) > p ( e )( z). (8.54) (8.53) - (8.54) e )( z) p ( e )( ). (8.55) ( z e )( z) ( e )( ) ( z z<e. 3 (8.53) (8.54) (8.5) e z max{ ( e )( z) ( e )( z)} p ( e z e ). (8.56) (8.48) - e e z e e 4 (8.54) - (8.55). e )( z) p ( e )( ) (8.57) ( z z >e
350 z p min{ ( e )( z) ( e )( )} (8.58) < z < (z p) (z p) (=3; =7; =) p (z=const) (=const)
351 {x = y } y- z=const - -. e z e 5 -. e e z e 3; z > e 4; z > - 5 e z e 4. e(xy) 35
352 p z a )( e ). ( (z p) ( ). - > > / - 4 a e e e (8.4) T= h = R =. - - y=3 5.) 35
353 -. - Mathsoft Mathcad (y=3)
354 8. 8. e z p ; =; (xy) (= 35) (= 35); =; (xy) (= ) (= ); =; (xy) (=3) (=3); =; (xy) (=5) (=5); =; (xy)
355 : ; ; ; ; ; - ; []. -.. g z ( A g = 98 ). - (z ). z - ; - z - z=z *
356 (- ) : ( [] ). 356
357 n M M M n - - (. 8.). = - (- ).. - : - G - R. : G m g m A sin t R a v (8.59) m v - (=n); A ; t. 357
358 - (=n) v v ( v v ) (8.6) p p b b x x x x b v x p x v - b ( Ox); v O v. x x...n (8.6) x. Ox: m x m g m A sint x...n. (8.6) x p O x. 8.. : t ~ x x ~ ~ dx x x A d A ~ ~ d x x x d A ~. m (8.6) ( - ): x z sin x (... n). (8.63) : 358
359 359 ) cos( )) ( exp( ) cos( j j p j x z z v x ))) ( exp( ( ) cos( j j p j x z v x (8.64) ) sin( ) sin( ) ( j j j x z p j x j v x j - ; = arctg (=n). (8.64) - j (8.6) j - j n n- (8.63) (- (8.64)) n- (8.6). : n 4n- ( ) n n (- ). n>..
360 -. : n - - [ r ]. v v arcsin. (8.65) r
361 - - u x x v x u v u x u x. - : v u u v v un u n vn v n. x u u ( u u )sin ( u u )cos v x x x x n n cos sin. (8.66) (8.65) /6 5 (3 ) ; < / : m m ( = 9); - ( = ); 36
362 ; z= g/(a ). ( +) ( +).» L - L nt T n = = T =. x n (t ) dt. 8.. L(z) 8. - L z- ; -. - : = 44; = 4; =. 36
363 «-» z =.... : : -.. :
364 9. -. [5 ] [ ] - [6 7]
365 [5]. -» (- ). ( ) [5 9 5]. 365
366 [9 9]
367 [8 ]
368 [5] : (5) (); (); (3); (4); (6)
369 []
370 A g A. [] [5]
371 . 9.. : :
372 : ; - ;
373 : ; 4- ;
374 B: (3. 9.7)
375 M J (. 9.8 A B) : - Z X Y. 375
376 : z A:
377 : z Y X. - Z X Y
378 «-8» (BY-8) BY-8 («-8»). 378
379 6 (. 9.). 9.3) : BY : BY
380
381 . 5 % : m m. 9.5 m m. 9.5 m m
382 - t o [3 8 8]. - m P. m m - m m : V mv mv (9.) M v v M M m ). ( m 9.6 m m : m m
383 m m : m m m : y W ( p) (9.) Q M p bp M b - p y - Q -. : 383
384 W ( p j). (9.3) ( M p ) ( b) 9.8 : - din : 3. - m
385 () [4 6 7] - () H ( t) Asin( t). - g g A arcsin( ) Smax (9.4) A g : g. (9.5) A (. 9.5) - : 385
386 mr A. (9.6) ( M ) ( b ) T T T T ] : [ A A T g g A A T g g. (9.7) ()
387 . ( ) ) (. 9.) i- t ]. [ i t i 387
388 . 9.. t ] [ i t i : g arcsin A (9.8) A m r ( M ) ( b ) (9.9) - A - M m
389 . 9.. (- ) ()
390 T. 5 5 %. - (. 9.3) :
391 : :
392 : - :
393 m
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