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 " ?3 N, ?' D( N) H A1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式）) Z3 }: R* }* W; \
3 P. f$ d+ b6 @) s# J1 m: C5 F
公式, R( _ ^. m' x% x
5 x3 j6 b/ M2 s
Where $ O/ y0 {* Y; Q& e 9 @9 J) \1 f( m8 {$ T" p, R2 a& P符号解释 5 {" C7 i( @# ]) F+ m$ F" _, M3 t5 Y4 @6 i, `! T1 f: F) [7 s H8 w9 g
According to the assumptions, at every junction we have （由于假设）4 Y! S& i6 B4 `0 A- O! h
5 l- D/ Y/ x, ]0 v( l7 y& B1 c
公式 ! E) P& ~6 S( t& K8 S2 X. I$ Y9 J% M1 {9 x) I2 O9 ]; q
由原因得到公式 ! H6 d9 ]) p& q; v! @7 X2．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）；& M B$ n1 |7 d' M
/ O& ^1 }/ w7 F, \+ l5 r9 N z
公式5 d3 u2 {+ t& I3 g) J# o- ?7 d9 }
0 D+ a+ L4 p6 [: F( o" |; }Since the fluid is incompressible(由于液体是不可压缩的), we have . t9 U: f. R: p# |8 O L/ y' H M: {' S# ~) U7 F/ r; E
公式' [0 {9 F& ~) A# R ~
$ V. j; a% b9 {" z8 u' ~9 |Where & n: D9 @' ?0 s' p6 Y/ x# b5 e9 h' ` ?3 m
公式 - B6 ]/ V; g, g m* i, Z/ L w, G9 ^- q
用原来的公式推出公式& l- _0 _" b3 R
3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到） 9 S5 z! ^ h n. }2 M; a " J) z: d* u- L* r7 A5 N# P3 g1 E公式+ E3 G, y: p# e, S/ F; W; d/ i- k
( q0 x3 L$ D& c% f/ B% F6 w. a/ a
11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：; C! C, Q3 a& o- Y/ J- L* H( Q
' e/ g( X6 U. B5 g) n0 i
公式 ) m- M- R' v4 p }3 R+ |8 v+ d1 H9 I, x) |
12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得） 2 f9 L( ~' ?$ V% Y7 _0 i/ D3 N5 y' p+ R' \7 r
公式 0 W6 `7 S, }; B5 r9 d4 z+ H9 Z
Putting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have6 }4 S/ l1 C! X+ R" F# Z
/ n" ~* h. C9 ^/ A! T0 C% r% k
公式 6 O* H& X4 ?% c! P" P : c/ X1 ]9 s: V: a; E3 t# ?# ~% GPutting these into (1) ,we get（把这些公式代入1中）: F7 w) }! G& A2 N/ g$ B
6 V7 c0 L3 {1 X, Y
公式! Q" L7 @: |9 C% ^% |1 E
& ^2 t. L: I2 n! B
Which means that the # h* L! b! t0 i, R: V4 L 4 n7 X& M% Y+ HCommonly, h is about4 b' h1 J3 J8 J8 m; G
! R- ^ i) J5 `4 Y, v, B- ?: }From these equations, （从这个公式中我们知道）we know that ………) ?7 p; _7 {1 B7 u- I/ e* q _
' }3 i4 f/ N/ F$ N& Z( M
3 @2 `7 I# @- ?, D+ ~
[/ N S. k' c- j/ K
引出约束条件0 z& I" |4 W* A, i3 }1 U
4．Using pressure and discharge data from Rain Bird 结果,8 Z/ j+ d- d, D9 C' D& n
+ a( V# e- P5 o& a- @- _: a8 {0 A
We find the attenuation factor （得到衰减因子，常数，系数） to be5 E3 N- Z! N& A/ f4 i* C
7 ~' R6 r/ p: ?" _/ z# [公式* n4 M$ G R$ A; `' e) j% w
; W' p# v% M0 _7 M5 [计算结果! y7 H; O) `% W
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）9 [; v+ s+ ?5 Y, Y3 V
3 x+ S+ s( J, B, |6 m% G" a
公式$ Y; U1 I* F% m( j* I4 ?
7 o$ F! h! U7 nWhere & }# A- l1 J0 u4 |5 ]% M 4 R4 }5 `7 C( E1 H5 L() is ;; $ E0 A1 Y; M" H0 g/ m/ g! c G% ]% ~
7．Solving for VN we obtain (公式的解) 7 @# y. ]/ H! v# U0 b1 B; K& l* N/ D/ h. S/ A5 |
公式$ g' Y; e8 N+ {% v' u
4 g; R! G( z/ \Where n is the ….. $ k( _' U! _' S2 _* O$ u- S" ^' s # C% Y; o- `% C: x , w3 R9 ^0 |% `! g$ X g
. q" E+ a5 v6 n ?4 U
8．We have the following differential equations for speeds in the x- and y- directions: " K: {# H9 U3 @5 n, W0 h$ D+ r5 _3 i( O- F% Y2 i# h
公式 ' \0 v1 W, n( s) p0 Q) L0 |% @4 X0 Q1 M, r% X
Whose solutions are （解）. r& @5 i: e% ~' K, X
7 T1 y# x9 F# p) i' I2 m
公式% H9 z. a, l2 Q% N1 M6 F' P
) t+ ~$ [' l8 s1 e- r9．We use the following initial conditions ( 使用初值 ) to determine the drag constant: I( w6 H0 R. @' g4 f. R" Q P/ X
9 ] \1 Z& k3 {2 W
公式% H* i( ?2 j: P
8 w. S2 W% P. r. ~) N* o根据原有公式$ b' L m7 Q, _0 a8 P- V
10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is+ I F( O# z5 p: |
$ r8 O/ K% O! _6 d$ w- q) t公式8 d% k' v4 N' p, v8 ~0 c c. }
5 c+ [8 V8 s# P" cThe decrease in potential energy is （势能的减少） : [) x& |+ U9 |9 p& ?7 l- U ) d% M6 ^8 D8 H( M; z/ I, c9 n E公式+ [$ T( p3 m0 V+ A6 t' N
! ?: H0 V9 J# _4 I8 tThe increase in kinetic energy is (动能的增加) ?' ~' \7 ]6 T4 Z& b2 Q- P7 @! n: Z
公式% X; H5 A8 }; s/ A8 I+ U7 e! r
, g0 T* A$ J& a6 B
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)* c- B+ c4 r' p" @# c
5 O: o$ x* o, m) y% EWhere a is the acceleration vector and m is mass 6 S3 V5 W4 D5 d& W8 j( g% |+ ?1 k3 F- S. m
! {. a% Q; i4 j+ W5 j' a1 u6 T4 M: e( |- A- _' L! z; d
Using the Newton's Second Law, we have that F/m=a and : x$ I( l- Y+ m/ J& B- @0 y5 ] % M- G7 W. P& H) R公式3 u; e0 _8 r. X$ P8 ~
+ y2 F3 G3 `% C- a: JSo that3 b* `! T4 G0 T3 F. E. X( B
9 K' ^1 Z3 ` ?8 k+ M4 ^) Z: [2 w
公式 9 Y+ x0 X2 y6 E F4 I! E( p, e+ ^$ ?
Setting the two expressions for t1/t2 equal and cross-multiplying gives ) a) M& z4 |/ U" | - m) l* w0 G) x. m公式 ' h- ^1 p3 a) n1 G . G: K, ]9 ?. a( v# `! d22．We approximate the binomial distribution of contenders with a normal distribution:' U8 t4 x9 } j( N$ m. ?7 A
5 S1 b e) A. b/ l公式! H; h% h' U) B; Y' t
7 d4 v6 g( @/ J0 D
Where x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives2 Z) H0 D% p; Y* D5 ~# H3 |, i: I
% g$ |6 q& W- J5 Y
公式6 Y3 A) K: m9 X! x! M
5 @! K6 \, @) X+ n: \5 h8 U& _+ bAs an analytic approximation to . for k=1, we get B=c& O- ]! W0 b7 S8 T" N
9 k" D/ B' `( V4 D, U- i3 {/ S' @4 W
5 j+ \6 ^$ e& I + p5 T7 J1 n; o26．Integrating, （使结合）we get PVT=constant, where1 e: J$ {$ N, [7 {6 ?
' T6 v$ s4 m) y' B* J公式6 y5 P& X. y9 l8 g
& R9 L# z4 w# b# c) h. j7 N% GThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so ' F+ t% g: _% Z! M/ V$ t: v9 M1 @9 a2 ?$ @4 u) g) d I% o" E
( C; J, o. U6 w
+ R0 s" i0 a7 {0 P" S1 X% z9 d0 c
23．According to First Law of Thermodynamics, we get& t2 x% [' M( E1 L. [
. w) O+ N8 O: J3 G' Y公式/ J1 [! I9 s- i5 Q6 @5 r4 ^
9 m" n" d- P6 t7 Q
Where ( ) . we also then have* p8 O ?$ _7 V6 Y( Y# X
1 N/ T/ m. y) x2 N
公式 + ~: L& S% A$ v7 P5 { p4 d3 c# p3 t7 B
Where P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:, @" m. V; S& N! B: C5 y* h4 _
+ Q, X' |: T9 |公式8 r! x: s! ^5 T; I( G
7 k. y* |, L" m0 r: @
Where, ]$ m9 F" p4 h) l' @4 ?5 l/ _9 g
" ^" e3 m/ U. I/ i0 R对公式变形! Y1 u6 k: R1 @, s
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到） - A5 }) Y3 b0 r/ ^' y8 H+ | , ^0 R$ l7 A6 Z+ O* K公式 7 w5 T p2 b0 ^) H7 O2 W# K9 F% Y# y; o: v! m. U# I
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 minimize0 _) {. G* f7 E
, D2 b6 z4 f& i3 {+ e( R公式: B2 w9 _* X3 ^
9 f9 _( w6 Z) e* P使服从约束条件+ ]) B5 D/ n1 w' H
14．Subject to the constraint （使服从约束条件） + d- H, u' w1 I. @# F7 s, q1 } p 0 B5 ^& a5 ?- S" o& s5 }' f, w$ C公式& x% }0 b0 E1 O" c* q- U5 g1 v
7 P4 X) Y1 C( \* g6 H6 I' e+ FWhere 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） ! [/ E7 }+ j" f9 u% _0 m. C: K) Y7 j7 S
公式 % o% S* g s7 g' p6 p. U/ d# P 1 e! L3 g" `5 A" `9 b% n6 |! [And thus f depends only on h , the function f is minimized at （求最小值）2 d( p1 t9 k: n: p
) D" \5 p3 \4 x& C, s8 |公式 + _, e% F3 e% T0 m3 s6 x 9 t6 S, i# \% z1 k; n* Q9 o8 K9 r9 ^At this value of h, the constraint reduces to ' q' k. G- I9 Q0 _- s% m) Y) O2 E' C$ @( K1 t x
公式7 e* e# e0 d4 i* V
) U5 f" V" p: X1 a: M) _3 W结果说明 / B, {5 ^, N. d. b15．This implies（暗示） that the harmonic mean of l and w should be7 c" v6 s# e) @) l
6 `: U# ]: P5 p$ v
公式! K: E# x N9 U5 |2 Z
6 m7 R& `% [+ M/ @So , in the optimal situation. ……… ' d) q# g+ I3 d* h. \4 v( f7 g V7 Z3 p/ T
5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is! K$ ^3 {/ H j$ Y# K- Z
& e5 U1 n' m8 E" \5 @" a, }# v
公式9 C8 w3 k& c& K& D3 ?! ]
8 z9 Q7 v( h/ i
16． We use a similar process to find the position of the droplet, resulting in5 |3 L# b I5 m+ |
( Z% D/ d2 P3 v- Y1 x公式0 x/ R9 ?$ f$ J* B) c& L: C i
, q( { S$ p$ `/ {, h3 g+ }) bWith t=0.0001 s, error from the approximation is virtually zero. % P0 E* F7 {2 U' a, U- C % r; A& w, i" T# w' _ ; j; j+ b( ^' W! E2 g& f& J- `- A' y4 i$ f* C
17．We calculated its trajectory(轨道) using : v) f- e# W; H- e & n5 w( J% E9 y% v T) I/ x公式 8 v" }8 G( Q7 J1 c- ` ' o1 o$ t; l0 l+ g9 A18．For that case, using the same expansion for e as above,$ b5 X& x5 O" a2 Q
* f. u* e4 K1 n2 t8 f公式 % A6 u+ M; Y5 f, W. x: P0 w$ P2 B. E: d% |# v
19．Solving for t and equating it to the earlier expression for t, we get$ _) M8 v3 r2 d. ?8 u
& y+ L( E8 A4 E i公式 3 A/ \- I- k' }- s0 W- O* q- Y: O0 U% z3 N, H) N
20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is " [3 c( ^" F- o; u0 O9 _* t) |" p) }, w- A
公式 5 o/ L5 M3 X. K % L# E, U; }# F8 h7 YAs v=…, this equation becomes singular (单数的).$ r) P4 U) F; X. R, V4 j% l4 x" B
. N9 m1 P4 v) ~7 V% W" l* D" P- Y ( M/ M/ C4 a" O4 o8 M d/ y ; j$ N% W3 }% T+ S由语句得到公式; z9 n9 j' a# t2 y
21．The revenue generated by the flight is ; I' g4 U1 |/ M2 O6 J8 ?8 h8 w3 V6 L3 [3 Z, Q1 [% ^
公式# X# `6 o% u1 F3 m1 H$ K/ D
" ?1 p" w, {+ ~0 |, p' w
3 Q; Y. b, v4 g7 I7 f, C; C, i$ s d- B $ j, ^2 f5 D3 M" Z3 X5 T24．Then we have( d- p$ m. v8 C( v
$ y. y$ r, D( G2 q, W5 q6 c公式 8 } ~5 R& e( v; a. `! ^7 x5 u3 I- b* `/ A M" U4 Z0 w& p( R4 | M
We differentiate the ideal-gas state equation 6 L7 L* y: y8 E- }$ m# l * v9 L% N2 W( N0 E# J- w& N4 i公式 3 s% I7 Q/ _$ K3 x" Y8 b, h- Q. ~2 r9 x3 L g. X% B
Getting 0 \7 Y% i8 ~6 G5 P 7 A8 d9 s1 W; c* F* ^公式7 y# K2 a/ V: |
0 D+ O! p" C1 y. I- s
25．We eliminate dT from the last two equations to get （排除因素得到）( N% M3 _! w$ d! ^7 O
; Z# l, D' B1 `$ u3 G5 G+ F2 I) u公式4 A0 f b2 s* t) O
( [* \- |8 i3 A8 X
/ \, |/ _, ?! D# h! C: Z% \! V1 @ I* t- x
22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations ) a; o! V- {9 ~8 e+ s: ]8 f, d, f, Q f X! A; b
公式 * S9 o" i: Q# f Z: [4 M. B' k! T) ]; g# ?
Where P is the relative pressure. We must first find the speed v1 of water at our source: （找初值） 9 @, `8 q9 D. x) }( M! b) J6 R. T: b0 Y" ^, d) b( ]: U. I
公式 : t% d& x7 l3 q———————————————— 5 c6 Q) g( z9 z版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。 ) y7 n! T8 T5 ]原文链接：https://blog.csdn.net/u011692048/article/details/77474386# u$ s9 P$ Z; m! @ 作者: 1369728843 时间: 2020-2-12 18:25
感谢+++++++++++++++; i- | P4 b( d; f" F 作者: chace 时间: 2020-2-17 15:19
学习学习学习 谢谢$ _- F5 o( v% \. M