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 ) W& E% _$ l$ H1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式）% i, H c( k8 Z6 J/ v, q% f+ k: q
, `( ^0 {5 w; t. b7 H4 L0 n
公式 8 t% Q; a6 S2 k+ ^- _' p5 Z1 @1 `' k$ ^7 A4 K7 Y1 ^% Q
Where 5 ]3 Z0 u) h3 i: F! u, l% h# [ s* [: [5 f0 V, y
符号解释 2 f/ N0 z) `/ o& w8 P: O+ z9 r( w0 y/ h
According to the assumptions, at every junction we have （由于假设） 4 Q2 X$ N5 y5 @1 B* O6 F0 d9 n . p5 i3 Y+ z2 }! U3 k- Q Q公式0 L8 I. B6 {6 V7 P+ q! e( [
8 l$ v8 d, u8 m) @7 M5 F
由原因得到公式# m/ J. D6 |1 ]- r8 w' p; p
2．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）； - \$ p, ^9 O( d; Y+ {6 R. ?: j# b. U/ S$ y6 K1 B
公式 l4 n; W3 [" N( e% Q1 ?! l8 R; l7 t! ^! d. Z
Since the fluid is incompressible(由于液体是不可压缩的), we have ) O y8 r! q7 J7 q# p 4 k" i" Y7 A q7 v公式; H2 L1 y2 K7 V& Q3 v* {
5 o I0 g' a/ l" N
Where9 N( U" C: d9 A# l' D/ o
# E4 y/ D- `4 |/ O8 g
公式1 z" U9 O+ s$ @) Q
) u' n# c8 d' a+ ~. ^用原来的公式推出公式3 q6 i: ^- r( `
3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到）, D. O4 P# h0 o
3 B; b& G9 e1 q8 S8 J) e1 E公式0 }( r3 H: v' _3 z
/ h) a! ?3 f5 R8 Y8 `
11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：9 W5 e% q4 i2 ?7 ^5 t
( D5 y! }9 v; s# i3 g; w, n
公式 & W: m7 q, L( ]5 j) y6 @- B' r Y9 `7 g: r4 n# ~
12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得） ! m/ l7 A G6 f; Y% k 8 O& j, i# \5 z1 C公式 ' X' f' W3 f/ \, U( g8 S# L# _+ m& K8 Q* ?' [
Putting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have: ~, A+ y! y7 T1 }: Y' X
( i5 u8 z: _4 `8 n
公式 ; [/ K# A0 X( p" B ! J, b! V! n! s4 I3 yPutting these into (1) ,we get（把这些公式代入1中）- j$ k' i- |' ?# p
$ \9 i/ {6 R2 o& ~" B公式 ( ~( p; G4 ^ \* {2 H9 J# e9 e3 ]& f& t & T& d1 ~: w7 `# p% i; cWhich means that the ; h7 n: w9 o- f2 Z6 j) L5 {1 [ # ^6 a+ s& q$ U, j8 VCommonly, h is about : _0 K# x! b; {: z) r! N+ `& `- y6 ?4 J5 I
From these equations, （从这个公式中我们知道）we know that ……… 1 {8 T+ J$ X+ l5 v/ w5 M 4 k+ @1 F4 O b7 L 5 V9 S* V1 B [$ K- h! M4 N5 T* a3 D6 c! ~# b
引出约束条件 ( S3 @( r4 Y) F. t0 v3 ^$ t: L4．Using pressure and discharge data from Rain Bird 结果, - H" L% Q, J2 ^8 h8 b z# p K2 a _; p+ RWe find the attenuation factor （得到衰减因子，常数，系数） to be+ c/ d, X" R; Z: A; i
& f; W) N0 J' ~$ G& Z8 ^* n
公式 2 ~6 z" Q2 o4 v $ w5 R5 z. G* r6 W6 G# V计算结果! D9 d' R5 \* A: B1 Q# @1 |
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） 0 U0 r4 @2 L; K0 _0 M : E. u. A6 @% f2 P' T公式. k9 s. P& U2 D* q8 ^
0 W5 S8 r, @) v$ D# }4 L
Where 2 y/ ~! }1 d2 S$ O, t' G+ b6 d* o0 z# W; K+ x: y. L1 U$ X( ?& x
() is ;;, X' M- O- r9 [$ i4 n# B& C
2 G( j6 K% C. D4 {5 n- i) h; N8 [7．Solving for VN we obtain (公式的解)$ t& Q) _- A. s
: H( ?1 ]- G0 w( ]: `( E: @公式! u% A+ |$ g" X( P
5 Y! M$ k$ O& w. Y) a; Z
Where n is the ….., F m X1 U; o1 A T, m( D- e
5 P$ s0 D2 S4 u, Y/ o w% R$ k9 m, A; j8 p$ @
+ X* Z" H' J; n2 Y+ W7 Y8．We have the following differential equations for speeds in the x- and y- directions:, B! b3 H# x9 c* R# F, J
" s, u* M g% X+ M6 X公式* f9 y) _5 i" |0 z
( n I0 P# Z- Y |Whose solutions are （解）/ Q' a, K' V" U- q9 d* p
8 c e- R. `' r9 w6 V7 c$ ~7 T, Z公式 . c1 D# Y! w/ S4 w* t 7 ~; c0 m: x+ X1 _. N; k: {. h9．We use the following initial conditions ( 使用初值 ) to determine the drag constant: 3 U8 u4 T. _. [0 N1 D' N + S+ Z! i/ z8 L) `5 }公式 6 t. _3 m% A- L$ `, W8 W8 L+ N3 z9 V/ {1 K
根据原有公式1 Q1 H G E/ W+ t' y5 E
10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is2 W# P- \7 E; E3 b& ]; [$ T, d$ q
1 Z/ J; R$ \; c% l* k; E3 ?7 ?公式. z9 @+ E! q) d7 T: e
' f: g8 {! B3 U; y7 ]& r1 x" W
The decrease in potential energy is （势能的减少） B9 V: K' E# c/ t
2 l, L; ]6 k4 u3 F& a& A! J公式 & S- ~8 K5 R4 b2 G* S" C6 U3 r4 V+ P: R. `' J
The increase in kinetic energy is (动能的增加) 2 ?9 B) _1 j. r+ l+ n9 B, h- v, p6 S8 f, [: B, M
公式# E4 Q0 M/ r& z- \( K1 r4 }
! P- S* b/ T* v* R7 J. i
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), y9 e" R$ D1 Q+ j; w! U
6 h( M0 ] {3 {, d G5 iWhere a is the acceleration vector and m is mass* `0 J+ M' @0 N: J* G0 Z
) _1 U/ Y3 n* s: e
* j8 X9 v4 d8 C% x% q8 r2 ~# y ?6 h# B: A' Y. {% V6 X
Using the Newton's Second Law, we have that F/m=a and 0 C6 U. b8 V: Y( ~- n& P - g {* \, c# C7 @公式7 ?% }0 V) J) o/ G
2 h0 a- H& ~) L, t! @So that s3 D; [$ [1 p5 y5 @+ _& y* s
2 }, o' a) _1 X/ _公式 - ^/ f1 `2 V) D2 G# B. H- C. ^% i2 y+ D: H$ D* [+ G% |
Setting the two expressions for t1/t2 equal and cross-multiplying gives : o ?) i6 o2 P& r7 X. o1 B 3 e# Q {. m; `公式 - s R3 M; f7 {- C8 f* }. L' J ! i5 o1 M }8 [. m( s22．We approximate the binomial distribution of contenders with a normal distribution: ( s# k6 G; l( N9 E3 E9 K2 E7 o3 Q( d7 k# q8 F
公式 u& V) O- d9 y0 r5 x, B0 Y
. G8 P# |3 e1 N, n: d. @7 f: \8 e
Where x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives 6 U! Z( K& W! \0 |8 x Z9 e$ @- O0 \# W
公式" p6 u, y+ U3 I
$ G# D* A% V) P( n
As an analytic approximation to . for k=1, we get B=c+ I. [. F& Q, M2 k: w
6 i7 q% r" r9 n K C ! J* K- X' @9 c
" N1 |2 r3 c1 V. [( u
26．Integrating, （使结合）we get PVT=constant, where ; b" t$ x9 N5 V( Z% z l+ D) }4 U/ O公式 ; I7 l K: |" J" }0 H/ Q" T + z' r# N7 s* H( QThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so ' _/ Q: a, l0 Q$ u3 t 8 y7 \2 y9 [1 ] Y$ e! m 0 Y4 J2 {4 @, G. i# x, F ^& {' [& ?5 [
23．According to First Law of Thermodynamics, we get! U q+ i) x2 u2 W9 u' \
3 d2 ~, d9 P' V8 N p8 V, i" b
公式0 z/ D) m# C- I; i' O) {0 Y2 d
0 [" G' T/ ~7 o
Where ( ) . we also then have0 s/ E9 `" s( ]' O
O( \4 F) ]! w& q+ Q1 `6 ~
公式 E) C% d4 d# j- I
- M" O5 x! Y( xWhere P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula: . Q, w: S/ J6 x' ~ " [4 |, O" i6 |* `( D0 o公式 ) _% U% A1 C) c; W1 u2 X8 M% V: s* i! T3 U9 G8 t1 D
Where 2 L" i; H2 \ m v" G: g 5 S% w) A0 N9 y 6 Z% i# {4 ]5 y9 h$ {
& A1 N3 D5 Y% t0 z对公式变形 : H. I0 s: [, G! l1 q+ w13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到） + z8 ]1 F: f2 \/ N. L m, c' F C3 Z* k* j1 r$ g1 Z
公式 1 i8 m' ? M6 p$ Z 7 d$ g8 O- T$ w) S5 RWe 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 minimize1 z2 P7 G0 x/ m( v1 f" ?& m9 S4 j
( P( T3 ]5 @ x H
公式+ k8 g+ l+ u. w! T0 |" v
8 W/ l1 s7 @4 R
使服从约束条件1 q' D6 r" p" A3 W. D
14．Subject to the constraint （使服从约束条件） - \; B, y; G B! f% O f9 C! P- J1 k0 n6 A2 y
公式5 B) }* x j8 C3 M% C/ p) F; B% |. _
4 l9 r( c0 l; ?$ I# _- _" G8 h+ E
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） . K. Y$ X, |5 u2 h- t' x3 C3 T9 h. J) ?# W
公式7 I: S M2 d. n" L
/ \; q! ] T1 P* r
And thus f depends only on h , the function f is minimized at （求最小值）+ @; F D+ N% Y! J5 t, b
( O6 N. {% W' k: Y公式 + N! w6 x S( U. o/ |* E# F$ I+ e6 z4 n' W2 ^
At this value of h, the constraint reduces to 8 w" v( I) N& G& U: @! i& O( X' T 4 a ?; h# u2 M; @公式 5 N. k8 Y* |" z: |7 | + w2 L* C% g2 a, H0 |; e结果说明 3 Q$ g! z) U) {2 t! Y" x15．This implies（暗示） that the harmonic mean of l and w should be - \( G4 i" I O4 h2 t% `+ |8 b. q* j3 K
公式 4 i; ^. o% R/ C' A! A- u! r/ N% A8 C
So , in the optimal situation. ……… 1 r8 D8 A- c5 a8 C7 R( w9 o 3 T9 O+ S0 t8 K3 q7 l5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is& r6 z) [) }+ I# `2 ?" L
$ L, I' z) U2 i r公式 6 k* l' x `# I6 E! q 2 h, t0 a+ y' ?) y* P. U$ a( [9 Q16． We use a similar process to find the position of the droplet, resulting in3 ]3 Q9 R7 X, g A, R
; |: X2 ^2 L7 i: ^7 G
公式 . Y5 O7 G4 }- O+ o + E3 ^' m5 U' z# N6 a, l& K2 uWith t=0.0001 s, error from the approximation is virtually zero. . n& q$ _: H( F4 \' j* Q7 ^4 Z8 s: Z( ~' {5 k# X: w$ r: q
( b) u0 K$ C- H) ]: Q1 I
0 R' \ M8 }7 l( V+ `9 r' B
17．We calculated its trajectory(轨道) using- u3 f! U$ G6 f/ X. C" \
6 d* [& h3 i6 q7 @9 S$ B公式 ' N* F# e* I9 Y( H5 x" F$ h8 t" |8 G0 y. W: w7 L
18．For that case, using the same expansion for e as above,4 O l6 q+ L; }: B
, {# O4 p3 }9 U* | s
公式 / f0 [& W$ l, ^' }3 T7 K6 ~3 @: g5 Q# ^" k* u4 i
19．Solving for t and equating it to the earlier expression for t, we get ( q0 W% Z9 ^( x' P3 ?2 t & k$ ?0 S3 y4 M1 N! S; \公式 ( [5 o% C& x$ h! ?) U+ X$ W# a5 ^) ]! O& e4 O
20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is 0 g7 n% e$ V ~/ z% }2 R* F2 z6 z2 C h
公式 2 |6 m: [$ _: I5 |- m9 S8 `7 _9 V
As v=…, this equation becomes singular (单数的).( T0 i# u7 g) `, W2 b
: |' M) I. H& P. G Z0 q 4 p6 Q/ R0 v& _/ {. I, o
8 ]. z3 J9 U/ F. r由语句得到公式) P% a/ ~; i8 |. J
21．The revenue generated by the flight is $ b9 B2 g9 W" u5 g7 {. t' |* U 9 Y2 Q- }( o' {( `' B4 Y" m0 h公式 : s# c$ X/ u K7 k: Z5 G: a4 n, M! {( K & Q/ e r1 y, |4 y/ q" v9 n ^6 L " l( m/ N, x5 z7 r3 R
% y3 B$ ?6 z( n/ U8 d
24．Then we have ) y5 ]& W0 W9 O* k$ ^3 T- A2 m6 D! s8 y$ x4 e& k" }9 |( }. D! t
公式: R5 I6 I$ \% S' E0 y( A3 G
8 E/ Q( p ^/ R' V) ^ n; jWe differentiate the ideal-gas state equation 6 V7 p* q$ h Y! o2 F$ m# F1 y1 Q" [8 l, n9 ~( }
公式 , y6 r& Q4 ~, ~% h& G: S; o " n( S, W, _ C, u1 pGetting ' \% B* L( _9 @1 q: U; U3 ~& l4 d9 s4 u2 z
公式 - @$ P1 [$ f9 J [& u+ Q ; h' q" V+ O6 B. g- k1 {: q _25．We eliminate dT from the last two equations to get （排除因素得到）: C- ?3 j: H+ J. s
# c: C5 z$ c3 d7 a9 K) M公式 ( z) o8 g# \) l9 n6 o" R. t* v8 R. x" N+ d( j# L7 ^+ B: t) L
% L) f5 b9 Z5 a' X- a+ ], i1 ?3 [
22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations6 u( ~& @( g2 {7 q, ~! [' L
8 i; c8 s2 `1 `$ Y/ ?
公式* H5 s8 W# o) ?& @$ c* b" r
- ^4 w: j1 R' }4 @: q, N8 p
Where P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）7 y- x: f8 v1 w3 b1 _6 f* K
9 ^7 [/ J4 |& l3 ~- W) t K
公式 / h1 V- b+ R# ^9 S———————————————— ! x9 o% X6 K$ f& ~% @版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。 ) I# d1 W+ m, C& q7 O1 p6 e, o( _) C原文链接：https://blog.csdn.net/u011692048/article/details/77474386 ) L c: b$ M1 l, r, Y- R8 U作者: 1369728843 时间: 2020-2-12 18:25
感谢+++++++++++++++ % U/ \. L2 Q1 c' x1 S9 V7 o7 K作者: chace 时间: 2020-2-17 15:19
学习学习学习 谢谢1 ]7 V- ?; k% M+ Y