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5 T8 Y. V8 ] @1 \ K# t: n/ {% n! d
1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式） $ n* d' ~$ n( j5 P0 E/ T: l/ d/ p7 _; n9 e! E; H
公式 2 u' L4 T# g8 f; s# |$ V2 w, w; l' d & a: a5 K! ?& x& O6 m! Z1 `+ p- nWhere " t, p( |, _2 \( C8 e/ H% ] # g/ p3 z3 m& T9 S& ^6 v符号解释 * b2 Q7 U0 f) e2 o , v% ]1 H) L' C/ yAccording to the assumptions, at every junction we have （由于假设） 5 U' w" Z; w+ y% F : o1 _) C5 X2 z$ u$ P, c" ]4 I1 B公式 + j2 m2 B. a0 x) n. `$ i% s; W2 x! X& Y- u7 ~% Z$ j
由原因得到公式0 x; R8 e0 O# \8 ? Q
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）；3 X/ w/ j. s& D, J/ }0 w
8 g9 X7 E; y! d$ c2 l: f) V
公式9 ^. a% \. c* R+ [
% G2 ~ {' \% p5 W. e
Since the fluid is incompressible(由于液体是不可压缩的), we have 3 C9 w J; S+ |: P) C# y3 {1 a) \4 C
公式4 j4 | v/ c- U9 N! N
; r: i" ~# n& s# m0 WWhere$ R4 Q: I* H) o( U& o
( P0 d4 F2 W. B7 a' h公式6 ]0 c8 W& z' Q( l% ~2 r
3 m! B9 M4 x5 V用原来的公式推出公式' v) l: \( i* [8 w" I& Z
3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到） ; `' t. z! f4 O' u: T( G3 T1 k% q; A. k: w- h0 U& X
公式, {, H% C* _) o% f. l, `
1 Z+ P: X7 H" G& M1 K11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：* K: u5 ^$ R9 Z! I N9 A( h
+ n/ E+ s8 M% i- V9 |2 C/ B公式. T5 c: A3 ], y; J8 Z
! s, D5 v. d& \$ q
12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得） 5 k4 N/ E% K. t+ P4 v / l ~3 o3 w% Q; s0 W" t5 y# H公式) k7 ~2 M7 x' c$ V5 U
1 M) U, b5 p3 |' oPutting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have; o4 K' ^! q4 ?7 _5 o/ {! `6 D8 A
2 ]; P# N8 d, X8 V1 x7 o公式3 o1 a. K `) }3 W
8 M& g z9 `& T+ K' I; W" ]& ]
Putting these into (1) ,we get（把这些公式代入1中） : \( Y) q; u* C ) n2 y6 J' U/ m J8 t# N8 M8 M公式 \. p5 M+ o1 ]6 b
" q, h9 }8 C& q0 o; eWhich means that the 5 T7 t1 Z. o L6 z( Q% t5 e( V+ \$ E. B9 ]
Commonly, h is about& m( ?* h& |! L0 y Y) `- _" {! e
0 ?. Z9 A9 c( z; }4 }! L2 NFrom these equations, （从这个公式中我们知道）we know that ………& q: Y4 M: G# u# h8 O P
$ O4 K& `- K8 H* }3 G , k* O2 _. W- }+ q! A! G& G% p+ k. @3 }8 M, M$ M0 F
引出约束条件 E: {$ K+ R8 I4 F0 @. h4．Using pressure and discharge data from Rain Bird 结果, ! i( X# H @% K7 V3 X& h& x# S3 A8 k
We find the attenuation factor （得到衰减因子，常数，系数） to be 4 ]$ a1 W, `0 ~: z) r1 r" K: `4 D6 D- [. s( P
公式# H$ a7 I2 [: i# u+ Y5 E
?' E1 S* b' D& u. Y4 a6 |
计算结果 ; z0 t! y. o! I) M6．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） ; O' s X% k/ e3 X / a% g: h; w# x) c公式! c7 P. T) _( P$ F* J, x2 [) L$ e
$ r5 N& G; H: {8 r3 P, V+ P2 _
Where * h' K6 T* O) s6 { 0 H. V( Q1 r" z Y4 A+ f() is ;;, D. r/ P1 @" i0 F% w& U& U
. S" x& R- u7 q% w1 q$ H0 w5 W
7．Solving for VN we obtain (公式的解) 8 [! S7 K( C% K* y+ x( x F0 U) Q4 s 3 r) f8 s+ y5 H6 [( Y6 F' K6 y公式 D+ s1 |0 p* v1 w3 ?* \4 o D( w/ \* H* Z( O q
Where n is the ….. : [7 p, ^ L* S3 F4 d9 }4 @, `9 C& e. ]' i% _$ @
' k/ A+ e! ?' V: ]" @5 S1 I# J
0 w a; F# i% O! y/ u& x3 t
8．We have the following differential equations for speeds in the x- and y- directions:5 C5 I8 k; l5 D5 W( S- M
6 |( k7 P5 W& A公式 2 y' z. J% l! V8 D8 Y* D) n" W3 d l
Whose solutions are （解） % O; p2 ]! T6 n \4 y2 `$ p( W % I2 S6 j, Y& R8 A" p% H公式 / z4 Y+ E1 f' O, K9 N ) E. o+ o+ F( z+ C1 C) w9．We use the following initial conditions ( 使用初值 ) to determine the drag constant: / b! {; o) V0 e4 ^1 A `* ~' ], h! s. e0 F8 X) U
公式+ x# s6 J+ B# i6 p
* f5 `& f) T4 y; y) {
根据原有公式2 @7 E( m# k+ x! K9 M
10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is & m% h+ U1 O2 o / g' ~' x3 q0 p8 p8 g公式 " n: J- O$ X: _3 _ 4 @$ t/ J1 H4 V+ Q1 WThe decrease in potential energy is （势能的减少） ( l0 d! [! q$ B- v1 b 3 K& m" u! |9 }; U公式8 `% c8 \4 c/ @& A7 M8 r/ g7 v
* g4 z: ^- g3 S
The increase in kinetic energy is (动能的增加) `5 E! I) w! n I h4 k9 m
: l, K3 b$ {: N7 u3 T$ \* D |3 o公式/ ]* `- w1 E* k- h( o% E
$ O0 k6 U8 v C7 ~2 `& ~. z% S% t+ P
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) 7 N8 s1 f7 i( Q2 R; A7 g1 o7 y. s- ?/ {. H! o4 U; c
Where a is the acceleration vector and m is mass8 r/ }4 `, H0 ]8 |( B# F
( v. u3 Y& W4 F# f6 G8 [ , t5 `- v5 ` k, {
* h: W' t! i( ^/ i; v
Using the Newton's Second Law, we have that F/m=a and; ~7 z! o. @5 b1 c
0 e+ G$ q) O! [. } h2 W公式 6 z+ |6 x: s. O8 N& C: Z' p7 g3 f" _- D7 X7 x
So that $ x% |# F6 e6 K# ?: ` * y. T8 d( w; U7 f0 C1 P+ f, C4 a% x公式5 ?- b8 C( Z, L$ d. h0 V, E
4 ?8 k: C# H' n4 u* USetting the two expressions for t1/t2 equal and cross-multiplying gives, B# Z( a# |4 `, h
) P4 y% q' n8 X) O公式 % L; i7 x3 p# p3 J8 G3 `* ?$ g0 K7 |: D9 A- y6 k0 e, K8 b N
22．We approximate the binomial distribution of contenders with a normal distribution:' m! E6 D1 @- R, |- D( }* N: _ u
: m7 q! _. Q8 o
公式 6 r: x6 P8 H6 P$ V2 @7 w: o" s* c& ~% Z1 F _2 Z
Where x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives 1 M7 A* w$ c. x9 i& W- [# w. G& a ) R2 z& H0 x" X4 h M3 C% [公式9 D: d# h6 O) `. h, i
5 }! Q/ t+ m; t/ n- X% `8 f' gAs an analytic approximation to . for k=1, we get B=c7 n6 y9 P9 u! h& U
7 [- E4 @4 v6 P; d/ v# v
7 E6 s0 P4 L& _4 Y, e
3 f5 o* {# z4 q& |/ v/ {0 g
26．Integrating, （使结合）we get PVT=constant, where' P7 X. D1 R- M5 U w8 n0 e K( T$ d
; F- p- `" [1 L# k; W7 K公式 y) L+ Z7 s# q * S$ l& s, R9 hThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so 8 g) W6 {" S! ^, D5 V R( I) b7 R5 V8 n& U9 n: K6 ^3 N0 [
3 {0 F9 Y$ N, a. l5 ]% E0 l0 v5 s3 x4 L) k
23．According to First Law of Thermodynamics, we get 5 v' b! N: ^' h ! A6 r" ^( l# d6 G公式 - s, }3 z+ v ]# N& u( P! z 7 u; T2 `; D1 ?Where ( ) . we also then have Y* ^- v% m# v& M" I& ], Q; m' k k1 c
公式 a$ h* b3 ^/ v- L: L: J( S$ J : s! r. {5 U' L; N& Z YWhere P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula: % C q2 R& C" R7 y* c 5 O6 N* u9 s# `* Q2 o( ?" Q公式 / x( V- {3 |0 E/ j * j A2 Y' x+ e6 W7 V- x& I$ R" bWhere ( ^' b1 v" G; D% P 4 H- h8 A' q* W * j" ?3 [7 w* G& R; q0 q: w
. ^6 a2 ^+ m3 E( z+ _对公式变形9 ~9 _7 d/ W& X5 @1 s0 ?
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到） 2 T8 r& V I5 ^/ R" R, j o, d' ]9 L4 x
公式 ' @; g8 ]3 n- J/ v! d! C+ k. O& q# ?) e9 ?) K$ W% W2 J( x
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 - W g7 [% F: J1 M - o1 h! @1 o1 T# j9 z& J公式 + G/ P" n4 w" ~9 P# w9 `. E4 c& [$ w+ O2 f6 M; o5 q" c
使服从约束条件 , u1 ~; x& |* \( X7 A14．Subject to the constraint （使服从约束条件） / D, y; `" \* z( | ) m% c$ [8 M, v- i# W公式 V6 X. f( y+ W % Q! {/ V' \4 a2 O0 T- ^" sWhere 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）. ?- w5 O# Q& e$ g4 O
, T* X8 ^9 W* M% m公式 9 f; q0 [# H ~9 m. g ! ~2 k# v0 [$ j; z1 w2 DAnd thus f depends only on h , the function f is minimized at （求最小值） # M* l* W2 t. E' t# s; _+ |8 h( R# m1 c$ A9 ?8 h2 Z* Q4 I! \
公式 ' h9 j3 f% O: F& e5 [! s- R9 m) |
At this value of h, the constraint reduces to/ u7 {- `8 n' @* \1 l5 i/ i
6 ?" x6 |+ i: v+ W! i# O5 r
公式0 n! ~( Z' u% \ d
2 d X8 Q! Z/ B, Q( R4 f
结果说明( ?: r! A# s2 t- ~" q
15．This implies（暗示） that the harmonic mean of l and w should be! C* V) U' t# T# v N
/ G; n2 D9 ^( ?" O* G, |, O5 s
公式* _+ B/ [' H, G% F' s U
7 y4 M; t; \. g: m, \; @So , in the optimal situation. ………' m; M5 g; F" q0 ^: I! j2 }
O; v+ D" U6 _0 k: W4 V9 m9 P
5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is : y0 P* w7 t3 r 9 y) r0 Y! x5 O% |公式 - }0 E% ^9 g- P0 @: ]6 | 1 B5 D% N r0 T" r16． We use a similar process to find the position of the droplet, resulting in : l1 R) t. x0 G0 [& K. W8 t3 W% c* ~5 b
公式8 ^: |7 S, e4 |6 J4 J4 j
. c0 ~' X$ c7 O+ `( Y; c5 O7 ]
With t=0.0001 s, error from the approximation is virtually zero.' J$ w+ H' R$ Q2 j2 F2 G
7 X# W# b4 \0 P' {4 X- m) b
, i D6 x3 {, a' p$ S" G5 M- { 3 s5 L" x& l! R( Z17．We calculated its trajectory(轨道) using, i" x; v v6 q' Y# |
" R, ]6 `* w# G9 R1 @0 K
公式; u- o7 M) d1 |8 m E+ G. g# L
( k8 \1 n4 q y( b, e( t
18．For that case, using the same expansion for e as above,$ }3 O2 u% G' T! M8 ?/ [% {% C
1 ~2 b! h, J; n) r
公式* }* b6 I2 n8 f7 {* S, x" _: |+ K; }
; b+ ~5 @% X" x6 D
19．Solving for t and equating it to the earlier expression for t, we get 3 {0 `. _4 d" V' `2 c2 L3 o. K* X
公式 * M/ d9 K. V0 f, T / g. J6 l3 u8 y, J3 c4 i( r& B20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is1 E* T( l: ?8 [) v6 c
; V' y, c# \- m公式 . w3 s, b/ I4 B* w. A5 V+ |; b ( ~. I% [! F3 D/ ~3 R. |As v=…, this equation becomes singular (单数的). 6 O& h2 c9 y( Z( G3 d. h 4 G& Q4 Z$ c2 w& u ! k# u2 E) {5 t3 o6 l ; ]. a( P; F' `: O由语句得到公式 + ^" h- @$ B- ~7 j5 V* u: R: m21．The revenue generated by the flight is 8 @' z$ J) M8 D; C9 u/ m: g) \) H/ l1 n3 @: O+ K
公式 " D; M4 X. [& U$ q) t , B. F$ Y" D8 \" u E5 ^2 s b, V/ r6 E6 V5 c
: Y) H$ {' B1 x9 k# i e2 `: B24．Then we have& P+ n. B7 N( T* z- }' Z* _; E
- W0 u- }5 p9 |
公式! U7 @8 x3 D* Q+ w3 c
% k% Z1 p! t9 S2 {7 ]1 U' lWe differentiate the ideal-gas state equation + N4 _: B* C; B# U+ {9 d/ C- V! m8 H. o3 b W3 q
公式 - P9 p2 l' x) X 0 n5 t P9 e5 Q" ZGetting9 G( m! n" T& c9 o5 l
P8 x" I9 U- O' Z9 [! S* J公式 5 s. F2 @8 O7 E, t1 P% A , o/ t" s- c* P6 l- G25．We eliminate dT from the last two equations to get （排除因素得到） ! j" D& U( m. i9 S7 C0 M( W * X: T1 l+ D/ S7 f3 i! H, t6 e公式$ j* \/ Y8 ]- E8 z1 Z, x
) _% D. h- q% v5 @( b+ m4 [ ) o0 f! Z5 K* \1 v* ?+ m0 _
* M4 e$ V1 I: _- _# ]) m6 C4 D
22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations + t; v* A9 J, J3 u R5 E6 {: M& M5 H0 b. i8 B
公式; m6 s# d( }1 D
7 R( v' p+ M" l6 N2 {6 p0 JWhere P is the relative pressure. We must first find the speed v1 of water at our source: （找初值） ; K, t5 u- J4 a$ d/ g3 m8 y- y' Z% S3 r$ R6 `
公式! V4 K) Q" V7 ~- b
————————————————2 F- m5 m7 L( b" t7 y* ~- S& k6 D, O
版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。 . j2 N8 B9 p4 `$ ~# o原文链接：https://blog.csdn.net/u011692048/article/details/774743868 i0 z7 w( Q% ]6 Z5 L3 l$ m3 t 作者: 1369728843 时间: 2020-2-12 18:25
感谢+++++++++++++++" ~9 ?$ b5 f" k' U 作者: chace 时间: 2020-2-17 15:19
学习学习学习 谢谢 7 q" j7 m* B: e6 T