This paper finally solves the problem of how to choose a suitable parachute when airdropping materials to the disaster area, so that all materials can be distributed on time and in quantity, and the cost of airdropping can be minimized. After reading and analyzing the problem in detail, we use mathematical induction to solve the problem and divide it into three steps.

Firstly, according to the data given by 300kg falling from 500m in the problem, the value of resistance coefficient K is determined.

Secondly, according to the value of resistance coefficient K, the maximum load capacity of various umbrellas is calculated by excel as follows: m 1 is 152kg, m2 is 237kg, m3 is 34 1kg, m4 is 465kg and m5 is 607kg.

Finally, the mathematical model is established by using the method of physical and nonlinear rules, and the practical problems are simplified, and it is concluded that six r3 umbrellas are selected under the condition of meeting the requirements of airdrop. The conclusion that the use cost is the lowest. The second question, let t 1 be the opening moment, and take it as the critical point. Finally, according to the formula of physical kinematics, the optimal opening time of t 1 is 6s. Through the above modeling steps, the topic is finally completed.

Key words: mathematical analysis, nonlinear programming, optimal design, argot, physics

Restatement of the problem

Usually, there are two issues to be considered when transporting goods by plane. On the one hand, all materials should be transported to the designated place as soon as possible and accurately to meet the needs of people's production and life. On the other hand, we should minimize the funds for delivery and optimize the allocation of resources. This topic focuses on solving another problem, that is, under the condition of meeting the requirements of airdrop, choosing a reasonable and suitable parachute to minimize the cost. The price of each parachute consists of three parts, and the cost of the parachute surface C 1 is determined by the radius r of the parachute, as shown in table 1. The rope cost C2 is determined by the total rope length and the unit price of 4 yuan/m; The fixed cost C3 is 200 yuan.

It is known that the material in the topic is * * * 2000kg, and the falling height is 500 m m, and the parachute is subjected to gravity and air resistance during the landing, which can be considered to be proportional to the product of the landing speed and the umbrella surface area. The umbrella surface is a hemispherical surface with a radius of R, and the load M connected by each rope with a length of l*** 16 is only located on the spherical surface directly below the center of the sphere.

The existing parachute with radius r=3m and load m=300kg is tested from the height of 500m, and the height x at each moment is measured to determine the drag coefficient k ... Finally, the load is required to be calculated according to the given area of each parachute surface, and finally the optimal solution of parachute selection scheme is determined by using the linear programming method to minimize the total cost.

The second question requires that the landing speed should not exceed 20m/s, and calculate the opening time of the umbrella.

Model hypothesis

1. Assume that the initial horizontal velocity of an object falling is zero.

2. The price of the umbrella cover has included the price of the rope, so it is not considered in the calculation.

3. Ignore the horizontal resistance of the umbrella when solving the problem.

4. It is assumed that the delivered items can be arbitrarily divided without restriction.

5, regardless of the quality of the parachute.

Symbolic description

Ri: the number of umbrellas with various radii;

Mi: maximum load of umbrellas with various radii;

K: resistance coefficient;

S: the area of the umbrella;

R: the radius of the umbrella

X: the height of the umbrella from the ground;

H: the descending height of the umbrella;

V: the speed when the umbrella falls;

C 1: umbrella cover fee;

C2: rope cost;

C3: Fixed fee;

L: the length of the rope.

Establishment and solution of the model

As shown in the figure, the force analysis of the umbrella can be obtained from Newton's second law: g-f = ma.

After deformation: a = (g-f)/m = (mg-ksv)/m;

According to the kinematic formula: a = dv/dt = (g-f)/m = (mg-kvs)/m;

The comprehensive formula is: ∫ dv = ∫ g-(ksv/m) dt;

V=mg/ks(C 1-e-kst/m) is obtained after integration.

According to the initial time t=0 and v=0, we can get C 1= 1.

Finally, the relationship between speed v and falling time t is obtained:

v=mg/ks( 1-e-tm/ks).①

Assuming that the falling distance of an object is h, the relationship between the falling distance h and the speed v is: DH/DT = V；;

Bring the expression of velocity v into the formula 1, and the integral on both sides is ∫ DH = ∫ m/ks (1-e-kst/m) dt;

De: H=mt/ks+(m/ks)2*e-kst/m +C2。

According to the initial conditions of t=0 and L=0, C2=-(m/ks)2 is obtained.

Finally, the relationship between the falling distance l and the falling time t is obtained:

H= mgt/ks+(m/ks)2*g* e-kst/m ②

Because the object falls from a height of 500m, there is a constraint condition H=500-x, and formulas ① and ② can be simplified. Finally:

v=mg/ks-(mg/ks)e- ks/t ③

x = 500-mgt/ks-(m/ks)2 * g * e-kst/m④

From the table given in the title, it can be concluded that the relationship between δ S and δ T is as follows:

Then draw the curve of δ s versus time t, as shown in the following figure:

As can be seen from the figure, 0-8s is a variable acceleration motion, and after 8s, it is approximately a uniform linear motion, so the average falling speed of the object can be obtained as V' = (53+55+53+59+55+52+53+54)/8/3 =17.67m/s.

When moving in a straight line at a uniform speed, the resistance is equal to gravity, that is, mg=f=kvS, and the data is substituted into m=300kg, g=9.8N/kg, and v' =17.67m/s.

Then from s = 2пr 2 = 56.5488㎡,

Finally get

K = mg/KVA =2.943.

Then, the maximum bearing capacity mmax of each canopy is determined by the resistance coefficient k, and the simplified model is added with b=m/ks, thus simplifying the ③ ④ formula:

v=b*g-b*g*e-t/b，

t=-b*ln[(bg-v)/bg] ⑤

H=bgt-b 2g 2e-t/b ⑥

Bring the expression of t in formula ⑤ into formula ⑤ and get:

x = 500+B2 * g * ln( 1-v/BG)+B2 * G2-b * g * v；

v=b*g/k/S-b*g*e-t/b，

The relationship between m and r can be obtained from the constraints of x=0 and v=20. After data processing, it is determined that the maximum load capacity of m 1 is 152kg, m2 is 237kg, m3 is 34 1kg, m4 is 465kg and m5 is 607kg.

Next, we need to calculate the price of each umbrella cover. According to the known conditions given in the question, it can be known that the relationship between the rope length L and the radius R is:

L= 1.4 14*r, so the price of each umbrella C2 =16 * 4 * L.

r

2

2.5

three

3.5

four

C2

18 1

226

27 1

3 17

362

Plus the fixed cost of each umbrella C3=200 yuan. The final total cost is as follows

r

2

2.5

three

3.5

four

C3

446

596

822

1 177

1562

The umbrella to be purchased now is determined by the maximum load and fixed cost of the umbrella. At this point, the problem is transformed into a simple linear programming problem. Formulas are listed by constraints,

Minimum cost: min = 446 * r1+596 * R2+822 * R3+1177 * R4+1562 * r5;

Constraints: m1* r1+m2 * R2+m3 * R3+M4 * R4+M5 * R5 ≥ 2000;

Additional: r 1, r2, r3, r4, R5 ∈ z+;

Solve with argot. In order to simplify the running time of lingo in the operation process, the upper and lower limits of the umbrella can be determined according to the topic.

That is, the upper limit is

2000/m5= 14，

The lower limit is

2000/m 1=4 .

Then lingo is used to solve the model, and finally it can be concluded that six parachutes with radius of 3 are needed.

Question 2: The best opening time is t 1. After consideration, we think that the best parachute opening time is to make the landing speed not exceed 20m/s, because this can not only ensure the safety of materials, but also put more materials under the same conditions and save costs.

Set t 1 sec to open the umbrella. At this time,

v 1=gt 1，s 1=gt 12

And a = dv/dt = (mg-kvs)/m;

The relationship between speed and time after opening the umbrella after integration;

v =[m * g-(m * g-k * s * g * t 1)* e-ks(t-t 1)/m]⑦；

Then the relationship between the descending height and time after opening the umbrella:

dH/dt = m * g-(m * g-g * k * S * t 1)* e-ks(t-t 1)/m/k/S，

You can get bilateral integrals.

h-g*t 1^2/2=mg(t-t 1)/k/s+m^2g-(m*g*k*s*t 1)*[e-ks(t-t 1)/m- 1]/k^2/s^2⑧；

Considering the critical state of parachute landing,

Bring H = 500m and v = 20m/s into ⑦ and ⑧. The data are shown in the following table.

t

y

t

y

t

y

1

- 137 1320 1

nine

-2855608

17

25374063

2

- 13306038

10

-276935

18

30 124246

three

- 12627435

1 1

2573 177. 1

19

35 145868

four

- 1 1677394

12

5694728

20

40438929

five

- 104559 15

13

90877 17.5

2 1

46003429

six

-8962996

14

12752 146

22

5 1839367

seven

-7 198639

15

166880 13

23

57946744

eight

-5 162843

16

208953 19

24

64325559

25

709758 14

The image drawn by Excel is shown in the figure below.

As can be seen from the figure, t 1 can be between 0≦t 1≦6 seconds, and as can be seen from the title, the landing speed v is less than or equal to 20 meters per second, so the final optimal parachute opening time is t1= 6 seconds.

Model evaluation

The process of solving modeling in this topic not only solves the current problems, but also has great reference significance for the same type of problems. Such as the problem of container loading. When solving a problem, we adopt the step-by-step method, which not only refines the problem, but also makes the problem get more comprehensive and richer results, which is a complete topic with more practical significance. The process of modeling is a perfect combination of theory and practice, and it is also a process of continuous exploration and progress. From this point of view, the model we established not only has a vital process to solve this problem, but also broadens our thinking and horizons for solving other problems in the future.

Of course, everything has its drawbacks. To some extent, blindly establishing models and algorithms also limits our creative thinking. Being unfamiliar with the use of various computer programs slows down the speed of solving problems and creates unnecessary trouble for the subsequent results.

I don't know if what you said is similar, but I guess it should be similar. If it is the problem of using a big umbrella or a bunch of small umbrellas, it can be solved similarly.