标题: 美赛数模论文之公式写作 [打印本页] 作者: zhangtt123 时间: 2020-2-12 17:14 标题: 美赛数模论文之公式写作 由假设得到公式 " t7 ~( I! E2 _ i1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式） ' V% T& ~2 g% \1 N2 X: y9 P' X' `; y$ e1 B
公式) r2 O' j7 _/ J) r: y. `
- Y0 {2 v* e% t! E' w+ r* x3 eWhere 7 D! a7 x9 `3 q- s! s' {6 i! c' @9 U9 m5 t; o7 ^% k
符号解释 / a! Y/ d0 T: C+ o 4 C8 c/ X' [; X, m: G& X1 @According to the assumptions, at every junction we have （由于假设） 4 X) L+ E) f/ X# H" l, ~8 l: p) ]! |: O4 _4 _# b7 }
公式, Q2 Y7 p$ u2 W2 I$ [' p& a8 ?/ l+ M
+ c; o: {9 ~/ B( ? c由原因得到公式 - s3 o8 r3 Q R6 T# K2．Because our field is flat, we have公式, so the height of our source relative to our sprinklers does not affect the exit speed v2 （由原因得到的公式）；1 y$ a' f: @3 ] E' P, y5 L
d% @# S; c" W2 n3 w4 ~* Z8 R t
公式/ f, P" C* M& O# F0 F1 M
7 o7 a; _/ d. N: X: x& o X0 r& ^; D
Since the fluid is incompressible(由于液体是不可压缩的), we have # m# e6 S$ T0 S+ P, e6 s+ y6 ?- t0 n' h% O
公式 6 Y; X( ] ^% l. p& ~/ S x* i4 F6 a; h G
Where , v$ r' ?" k6 z" ?9 R9 T5 E # g7 v. ]1 }* J公式 1 Y7 V. j/ n7 W4 r; i/ |8 ]1 [( z9 F2 b* M6 _/ V4 S( c0 p
用原来的公式推出公式 / n# J3 p0 @1 S }/ |) W3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到） 9 E: Q* {" z( N4 Y' s8 {( }, a! F. T. g3 n
公式 O) h& i3 x( c6 O; J+ f" A
' ~( r3 }1 D, \- U: i
11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：1 ?( t: R! F {4 C( | M3 l: c
2 `4 `% G8 n( I3 y/ B
公式 4 h: d6 y$ j* r8 T/ b* N' |1 P + B6 L/ a( r d12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）/ ?' a" ~- m1 @- B2 |- s" z7 W$ a
; G! g. l$ O, r* ~" n( r' ~2 i" k
公式 * v3 r8 F3 _7 e% _) d% c* b" u, a, M( I8 `3 ^' m
Putting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have8 o5 N# \$ i5 P e
% ?. }$ |3 Y- w- v
公式7 X2 \3 I4 k9 y$ Y- C, g
# Y! L! s1 O6 H" Z- M4 h/ ~' |Putting these into (1) ,we get（把这些公式代入1中） ; O0 {6 Y- P# Y6 x1 B1 p" h f5 u! [/ x; G; ^/ {
公式6 [' |+ r; [& z8 z6 z8 W
0 w4 W! o+ w% h3 L2 Z
Which means that the # b( G# i* f4 E9 Z0 q# p. D, S# y ]& C v: x9 J
Commonly, h is about ; V& x) S2 T8 }% Q, |# v( P! @& V% h7 f . V, {8 K% j. G4 PFrom these equations, （从这个公式中我们知道）we know that ………* b3 Q) V2 [0 d% D3 x# w, _3 Z
6 M8 l' v1 m6 Z; F9 y
* ^0 D' W$ u2 S7 N! L5 ?. C3 @
, }4 [4 d7 l, {4 N引出约束条件 * [- @( j) k2 k1 d3 B' s% l4．Using pressure and discharge data from Rain Bird 结果, / n9 T6 z! J% m- | 2 b' x5 @4 |8 H: d: t( d% K* OWe find the attenuation factor （得到衰减因子，常数，系数） to be % M/ c) v, _" i" `3 \ # Q y/ W" Z" ^% J8 Q公式 + o5 _' X1 V2 Z6 ^7 a7 j! X; n2 @$ V4 ?, ?% j, q. s" l0 a2 s& d; M
计算结果 % W Z6 K$ W7 J* r2 `6．To find the new pressure ,we use the ( 0 0),which states that the volume of water flowing in equals the volume of water flowing out : （为了找到新值，我们用什么方程） - d6 c5 h3 H2 i$ K+ Q( i3 |5 D! \) l1 O% C
公式 7 m* L9 O) q6 d% K) \8 [ / {; J: r- V1 f7 QWhere 7 I+ p( L+ z+ A, @7 U/ \# m- w/ Q+ H6 x3 o$ N
() is ;; 0 b' n" w+ n" ^* F ) R9 x* C, m. i! c7．Solving for VN we obtain (公式的解) 8 x" P. t, N- n& b1 p; T+ g9 z; u' L, U. y+ A5 u1 m& b7 |- E; \) L
公式, e7 q1 n4 p4 ]
4 w6 z4 F8 `2 }* z+ d, s
Where n is the ….. + s9 u' p0 b! f; T1 K# l7 H) I; }6 r$ q( f1 y* `& n ?
( e. Y r- E" k; D6 L7 d7 l+ f) Z
; i$ S ~) U( }' P. a
8．We have the following differential equations for speeds in the x- and y- directions:4 `2 d7 L$ g* l8 j9 f/ R, @
4 g8 V% i w( C. }Whose solutions are （解） . w; l' \+ W# E + p' R6 h- E( T$ s( H公式 % n9 R6 o0 L' E# W" u" y, s% p. c) Q; X
9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:( F( F5 K3 n o& a4 }
F' P6 K, X0 e; \公式 , l, q# z8 t ]; q' k& T% S) t$ r7 X4 |+ t7 R+ n/ N
根据原有公式 3 ~9 @: ~* U5 X; m/ u10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is 0 O) D6 d# h. ~* ~4 H, `' e2 k% y! _
公式" h7 e0 k% T) B6 W6 n# Z
/ C0 p5 N9 g" h, q+ j/ ^1 h
The decrease in potential energy is （势能的减少） , X+ a, a5 R& l; R/ C/ n0 J, z' e6 y- ^ @
公式 ! a+ `- O& ?( W: Y6 ~# l( \6 }# s* ^" ]$ V0 z
The increase in kinetic energy is (动能的增加)" ^1 G& ] j- s2 {9 F2 T& h$ A
; I8 N& O. r8 j
公式 T# }; j5 p$ O& A% S! x1 b$ u# f
Drug acts directly against velocity, so the acceleration vector from drag can be found Newton's law F=ma as : (牛顿第二定律) ! r( p, |% b) K# s6 g % k: B6 t0 n, X. s1 tWhere a is the acceleration vector and m is mass - l8 c% @- R$ G$ H, b1 d 3 G0 @4 T ^6 Q; n# M! G o: H3 B# w, Z" F! e% b6 ?" H" Q* [6 K/ x- Z: x1 g' k
Using the Newton's Second Law, we have that F/m=a and: l, r- [7 ~: c
9 w. A5 D0 D; R1 V1 h8 C0 |) o }
公式 / X- b3 z9 [- r7 J% P - L8 o1 ]3 S3 V- x2 p, `+ rSo that4 g4 @% K( u3 W1 X# Y) r
# ~$ d0 y# n( F' ~% _1 `; c" \公式) D1 @" r3 F e3 ~# p4 X; K
6 V% G c! T. t1 C: t3 [Setting the two expressions for t1/t2 equal and cross-multiplying gives8 B/ R8 Q4 `4 u# Q
( S7 |- Q% n9 r' q6 @
公式 ! M' \5 |" z4 S# a) Q 4 J; `, N3 n' J& G6 W22．We approximate the binomial distribution of contenders with a normal distribution: : y) c# F* Q- Q4 ]+ @1 n8 z- i7 s% W- t, b% A
公式( s% J7 j/ j1 @# K/ H" W7 @2 m
2 b8 V( R+ G: uWhere x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives & r1 F8 e% u( ~+ G5 K) M$ A4 F * r3 w( ?' E% ~5 T0 B2 ?公式! p' i2 F9 ]' i- ]6 \# y6 ~: z
( g) k8 H# ~! {: d7 f5 M
As an analytic approximation to . for k=1, we get B=c . E& n& v% B! E( G9 ^+ O* M+ S4 g/ q
# ~' O9 g. }0 u5 t! S0 g
8 m2 N* g& a! C5 A1 `7 i% S( |
26．Integrating, （使结合）we get PVT=constant, where% n" E9 e* x3 p! D* L
# w; P2 V4 @; a% O# r. H$ e
公式% E( ]7 \6 z1 d# S& n7 V
9 r5 d$ \5 U- F; iThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so4 q8 O( |- A. z4 V& I. X9 R0 d& ]
6 H7 \. c) h# z( L: Z8 g * D/ z5 l N$ v: \
3 }" a% W/ I" a
23．According to First Law of Thermodynamics, we get , @( f' A! {; p, L0 G' e( \+ O 9 w; Q1 o, L5 j. L3 {/ y公式 / x6 q& X( _. Y( {3 |: U% f$ u, Y5 T/ o6 {, w
Where ( ) . we also then have4 i# k( D8 E7 y
% M* u0 T$ h3 R! ], ?公式 ' R/ d; K/ p, Q/ G: F) h+ O5 p; x7 M: B( I. G
Where P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:! T, x$ A- T$ R H1 ?2 |' R
3 t5 x1 }3 D! E1 I' y
公式 % ~3 _/ {0 o+ L1 w3 D4 L8 a) C, P 9 t' C/ T* S7 I( U% o! |3 ?& QWhere 1 |1 E% B2 S7 a; ~* _7 k , [% w9 t P1 S7 W0 j9 f: z K' F) E* G/ p3 k C ?2 t
3 X" m, S( ~# Y' _% u5 n
对公式变形 ' W8 x, Q9 y2 m13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到） 0 A+ S/ i2 t O' r! |# z1 Y* n6 c6 r; ^* B3 k; W5 R w8 B
公式1 S- N% c9 M1 H. p P6 g, F! n
$ N/ [7 J! b6 ^( u
We maximize E for each layer, subject to the constraint (2). The calculations are easier if we minimize 1/E.（为了得到最大值，求他倒数的最小值） Neglecting constant factors （忽略常数）, we minimize. O1 d' I0 U9 F+ x; z* n
5 }( W' F2 O f% E6 z Q0 q* e公式 6 X" ]+ T: y: I4 M- a6 k. r6 Y! W9 |- q/ l
使服从约束条件3 K4 {8 @1 ?* L( z( s% v2 ]
14．Subject to the constraint （使服从约束条件）8 \; y" {& z" w6 U/ K6 I
) F- ~: g4 m3 `: |
公式- E1 q0 A( N4 L; j# C
1 { l" w0 }0 L
Where B is constant defined in (2). However, as long as we are obeying this constraint, we can write （根据约束条件我们得到）* w c; `! H3 U! t0 Y5 D( _0 P
% \9 X' \+ A1 S9 p( S8 P" G- v公式 0 F( U# ?3 J; g# K8 C( y8 Z3 j% P2 o$ k6 g6 I, t2 y
And thus f depends only on h , the function f is minimized at （求最小值）' ~# c$ m" ]9 n: D4 B7 s+ k1 O4 J
+ [6 N K+ `! e! F& h: {, H4 w公式 ; U6 n1 T8 b4 u$ D1 B 8 e$ J' \6 F( SAt this value of h, the constraint reduces to ! G; V, s) D: _ V3 O$ ~! k) e! X ; F6 w- T5 e/ X8 X9 y T, I公式/ f( v. o8 z( m/ _( f1 |
: U* j$ u* { E% i结果说明1 v! N$ b' t4 z
15．This implies（暗示） that the harmonic mean of l and w should be# x* D% q E- r& V8 k
0 S- B; ~ }" i+ j
公式3 l7 P |: k9 J' k0 f6 B" k
* c' X- c1 h; t" \+ H0 z% P. h# U' YSo , in the optimal situation. ……… 2 y" s- w8 a$ P0 ~. X1 e$ B3 N 7 e% n7 @1 {1 I& u# g3 Q5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is 5 w$ U+ f6 r$ k% L1 n8 z' k9 a3 o3 ~. t8 Y+ a8 d2 N
公式# k- `! f( ^! x
& E4 J `6 G' P4 N16． We use a similar process to find the position of the droplet, resulting in& q7 b* r: J2 ?. K; A$ H
- f+ V* Z5 s" Z# i% K. _# B公式- x, k! w5 d/ `2 D" p8 @
( n. w$ K- Y5 c3 T9 S9 A7 g8 N
With t=0.0001 s, error from the approximation is virtually zero." q, v" Q9 A# h
) W% }3 } P% N/ [
$ z( \( W* c( H0 E) U- z* G0 B/ M
/ F. v" x4 T# |4 b
17．We calculated its trajectory(轨道) using . a8 B* P Y4 w8 R" S7 W0 D! m9 B6 q4 {# p- a6 }
公式 4 Z8 E0 R1 W6 {2 u8 [& F8 i' t+ P " s! v M1 [4 C18．For that case, using the same expansion for e as above, v8 P4 y9 Z( r# h" H# P: c # V" x4 }3 z' F2 K1 Q, u, A公式3 x( M A2 \, I6 K6 S$ P* @
) Y5 p1 f! f% ]1 _% z7 U19．Solving for t and equating it to the earlier expression for t, we get* a9 K4 y# b) v/ g3 W& z
3 l" j& S2 A7 q+ Z3 x公式 e& Q% [+ {; N. V2 R$ q# ~
e a4 |7 m; c: J) ?( K20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is8 l- q% ~0 F( l, R% g& l/ i
5 R) p/ A4 Z+ M$ U$ c
公式7 z# l- t1 P9 A0 o- E J0 T3 u, }3 m
+ A; x. p& o+ o$ ]As v=…, this equation becomes singular (单数的).( F% \1 i/ _, E) F
/ f: a+ Q6 f* S9 W ) j9 k7 D7 w3 @, Y+ B' D0 `& j0 J" u Q, X) G7 v4 r. V
由语句得到公式$ y) ]: ` K- B% }
21．The revenue generated by the flight is ; m; E4 h2 i4 o' V& \ I1 N# O1 ] R3 Z) a* H, T: T
公式 " g4 @# ~1 t# Y) y/ D* ~ . k5 I1 z; z! ] ( ?5 Y* `( ]# J1 E! F0 \* d
& E3 w$ W& i4 G- V# x% |24．Then we have 2 G. [3 [( ~1 t3 V; a3 e. L' }2 E$ J( ^' K( N$ v' ~2 p- G' j
公式6 X8 x/ Z( ~& w4 Q- R; f0 M
7 q4 i" L+ _1 U+ ~+ P
We differentiate the ideal-gas state equation4 q9 e* ?& G- ^
1 K6 v; }/ x" v2 x( R公式 5 h# _- f ^9 E( A. N8 W( U 7 A! v. a$ x& P. H$ R$ v" uGetting4 V- E" [' A1 }. T% U! w$ t3 O
2 w5 m* L5 h9 N. l/ Y1 j! W公式5 k4 L" f4 }" L, W L
! \6 `! H4 k1 s0 I. n8 [& H25．We eliminate dT from the last two equations to get （排除因素得到） 9 _2 L0 P7 O8 y $ O* f$ r# U% B ]. P% J公式 8 F W5 [6 {+ b- I; X 5 Q7 s2 r( h3 F# @ 9 B6 @7 ]: F/ L# J7 S
& u# b6 T% ^ B+ N. ]) l22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations! b! V3 l' L: q; p0 {
4 m1 a! k' D/ B& J& Z
公式. J$ x% Z3 t, o- q" j
1 A3 r/ K* G; `7 R: {! x
Where P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）; r( L' q, }; z* \8 u" i
" O5 }3 i' H, i' c$ v' o; P8 O
公式, _: \, m# h: l5 [. g, A! ~* D, [: a
———————————————— ; K* w h! }, ?1 T" O版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。 7 p5 R n* y Y. x" G原文链接：https://blog.csdn.net/u011692048/article/details/77474386 2 t% `# c N! H( c/ v8 ^# g$ ?; `作者: 1369728843 时间: 2020-2-12 18:25
感谢+++++++++++++++; z) a( h. a( i/ [) b$ b 作者: chace 时间: 2020-2-17 15:19
学习学习学习 谢谢3 F, n* C J2 b