美赛数模论文之公式写作

zhangtt123

由假设得到公式
1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式）
0 f0 l7 J" c1 Q/ |
公式

Where
( Z( P' N! N; ?& @1 h
符号解释
4 }0 g& X0 P+ L
) B! q7 [% ^( m- W/ tAccording to the assumptions, at every junction we have （由于假设）
% f8 s8 h8 X$ G
1 Q9 n  n$ p# t& n: r公式

2 a, E3 }* l8 F5 J- S" o由原因得到公式
7 z# H1 J% i7 F4 Y6 C; a% U2．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）；
8 t3 Q* f8 s9 B/ ]& y
7 c# g# ~* Y$ k/ b+ `# m* u+ P) I公式

Since the fluid is incompressible(由于液体是不可压缩的), we have

- H& X. T7 c& ?6 S  `公式

# _% P6 x  A7 ]7 ~Where
) o$ W, Z/ v# X. n! J; e, c
# ?* A3 K# [$ e5 h3 x% B8 u公式

! D- B( N0 l: X用原来的公式推出公式
3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到）
; b9 [# r, ?% V# K7 W) h5 N0 g
2 G, |9 e8 G0 y* K& T公式
1 d' _. N' \3 w! l* u
3 t% [! Z5 y  G% q11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：

公式
! z' W  j% T7 \* j
' J# j7 K" W. A8 F8 _12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）

公式
2 ~, s6 J, J+ n5 W
Putting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have

公式
2 ?7 H5 \8 L# h( j8 g5 X
* C  S2 V+ x+ `: f2 N: NPutting these into (1) ,we get（把这些公式代入1中）

公式
8 b1 P& c1 P" N' p, H) Z
) F& w% c8 Q# T* D  IWhich means that the

Commonly, h is about

! M3 G; Y6 j1 H% dFrom these equations, （从这个公式中我们知道）we know that ………
8 i/ x/ ?* y3 ~! T+ D$ l
# ^0 ]3 A  ~  Z8 v 

+ M: y3 Z& l+ ?, ~) S引出约束条件
4．Using pressure and discharge data from Rain Bird 结果,
4 t, j( r0 d9 p8 L* r. Z
We find the attenuation factor （得到衰减因子，常数，系数） to be

* \# v- h+ l  x6 a6 M2 J公式

5 j, d1 I+ }; G% F* }$ O% }) g计算结果
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）

公式

# Q' t  x6 u/ q* {  Y- Y  x" XWhere
5 c; T3 p1 ~0 ~$ r4 B% N1 j6 z
+ v9 P: ~, U4 j7 G2 e. B! A+ l() is ;;
% o5 \% w- A/ Y: S% O
: x1 H& M$ |  p: q7 Q7．Solving for VN we obtain (公式的解)

公式
4 L( V" X. T' C$ Q8 \
Where n is the …..
1 u3 O) w( O$ w/ H/ \& H$ n
 
- @6 {0 W7 Y( q7 H( O
8．We have the following differential equations for speeds in the x- and y- directions:
1 [; y; f8 m0 k1 n: G' o  K
公式

  |) c1 Q) C5 P8 F3 C1 c, MWhose solutions are （解）

公式

9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:
7 b% j$ o! @/ T# _# y
公式

8 l( J; M3 e; q9 h$ j8 x( R根据原有公式
+ i3 e, |9 v, {6 v( g10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is

公式
* q( r- F: H, ~3 l* T; o4 a& f/ T
The decrease in potential energy is （势能的减少）
- N& u3 v) _3 k0 e
公式

The increase in kinetic energy is (动能的增加)

公式

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)

. N$ m. N, P- \Where a is the acceleration vector and m is mass

 
* @" r# ?& k" J! {4 t6 |3 h
Using the Newton's Second Law, we have that F/m=a and
: U& x6 }+ G* X9 M5 E% F8 Z
# l0 V; R! ?6 j$ Z6 \$ D公式
! K: K1 d+ M* j" ^1 u
So that

公式

Setting the two expressions for t1/t2 equal and cross-multiplying gives

5 @* w" m9 x& ^2 b  s8 Q公式

& {# u7 z- d6 }1 V22．We approximate the binomial distribution of contenders with a normal distribution:

5 G9 f: u- m5 C+ d* J5 k$ G- |$ O公式

Where x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives
0 ^  C8 z% f' a% T# d" F- M
2 e4 O/ f& m; b2 r" Q公式

0 x' x! x$ X) {- b1 A7 FAs an analytic approximation to . for k=1, we get B=c
8 [% d+ G7 J! t) k7 z  V
 

26．Integrating, （使结合）we get PVT=constant, where

公式

3 f5 r" p  g% i% F* Y2 p. s- _The main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so

 

23．According to First Law of Thermodynamics, we get
! N" j: n; |# s' Z
公式
0 J/ D) T  W+ N( z
Where ( ) . we also then have
6 I3 p" A0 q. o
: Y5 q; l) }/ W: e' i+ P6 \; n: I. j公式
# k  T% J; t! U9 ?& G) e% t& W! D
Where P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:
( e4 L, D1 F1 J( X; _! g; E9 r
公式

Where
1 B' f: ^: F4 h- i: m
) ?! o" r' L. F: K4 | 

- [4 n% Y( U8 s6 _0 A对公式变形
) E4 d; f" o# K2 p$ g8 x! y, J; Y, X13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到）
* Z% {" }* b9 u# j7 \  N
公式
% P! ~/ x4 L) G, H
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
4 I/ ?9 |2 t  @) ~% q5 p
! i2 E# Z4 q0 p1 Z公式

3 V; H. U" Q* s8 j$ @/ G# I# d使服从约束条件
14．Subject to the constraint （使服从约束条件）

公式

& Z4 V; X4 r6 |0 RWhere 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）

* ^' ^2 d8 g. n$ b公式
# W. `6 H, A) j! g- A
& o% y7 Q' d  f9 E0 V) PAnd thus f depends only on h , the function f is minimized at （求最小值）
  r/ O1 I2 F8 n& {+ S5 X
, N  _: m1 H9 c( Q4 p" k4 C公式
, Z# W1 y2 q& x' C; L! @
At this value of h, the constraint reduces to
2 N' o' S9 m8 ]) G) ]
/ ^% E2 w- W# _: e+ I4 u, O公式

; w6 c# L5 T: b0 |3 i结果说明
15．This implies（暗示） that the harmonic mean of l and w should be

公式

  q# g) n3 N1 L$ ?So , in the optimal situation. ………

; @3 M  y& m* T4 n5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is

: Z5 u' V8 }" A7 A  B7 z8 H, R公式

16． We use a similar process to find the position of the droplet, resulting in
9 Y8 [3 p" z7 _% K, _
7 Z' V" {, |& B& H$ s6 o8 f1 f+ h公式
9 [/ _3 V! L1 v' G0 u8 A
With t=0.0001 s, error from the approximation is virtually zero.

, W& e, u4 _" b  Z, D( \! h  P8 I 
1 c$ b8 K7 @/ z& c. y4 B, x
& v. q4 Y4 A" b5 n  Q17．We calculated its trajectory(轨道) using

公式

18．For that case, using the same expansion for e as above,
$ E9 j& Z" S' j; z. \4 ^
公式
/ z) F: }& s  N" V& j/ y" i  ]4 a8 `
, D. e" B( X* `% h+ M/ `% t  T" ^19．Solving for t and equating it to the earlier expression for t, we get
0 R5 r! N) w9 L6 s: v/ L" r, |$ i
公式
$ \+ \7 o6 `) S6 \3 D
% n. r3 i" E  w+ y$ `20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is
" |( o$ g! a7 a) x
公式
8 ~+ k, ~9 b( y" x  r4 p
+ N  o3 x4 |+ E' lAs v=…, this equation becomes singular (单数的).
- z( ]4 v7 z# u% b& d) j$ _
 
* j* a/ U; c/ H2 G
+ k# z- L/ d" n) p由语句得到公式
5 s5 V' T9 e, e9 ?21．The revenue generated by the flight is

公式
2 {6 i, E  k' Z. @  @, X/ z# t
* p$ b1 L* L8 { 

24．Then we have
9 W1 j5 r: ?2 c+ ^3 o
  ~- d2 A3 [9 ]+ w  i公式
# @7 m- f* {# b# |' h" u
4 u- ]7 I1 k% w9 jWe differentiate the ideal-gas state equation
) `8 a! v6 r: B. \, n$ e8 d
公式

Getting

4 p9 i0 E6 I- Q公式
# [" c' m3 f5 G# \4 l9 [
25．We eliminate dT from the last two equations to get （排除因素得到）

! h( N, k. Z: C) ~' v! K6 `公式

 

22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations
3 W* c$ M6 Z6 V& k
公式
1 f0 u$ G- O9 r5 m/ ^- _3 j4 h8 D
9 _) }- v' }* a8 M( b1 O$ iWhere P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）

公式
————————————————
% M$ ^8 v- }& X: h版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
原文链接：https://blog.csdn.net/u011692048/article/details/77474386

1369728843

感谢+++++++++++++++
' i+ }, c4 j- b- q

chace

学习学习学习 谢谢
% z. b4 `$ \' W. ^9 Y