1、第四屆“認(rèn)證杯”數(shù)學(xué)中國(guó)數(shù)學(xué)建模國(guó)際賽承諾書我們仔細(xì)閱讀了第四屆“認(rèn)證杯”數(shù)學(xué)中國(guó)數(shù)學(xué)建模國(guó)際賽的競(jìng)賽規(guī)則。我們完全明白，在競(jìng)賽開始后參賽隊(duì)員不能以任何方式（包括電話、電子郵件、網(wǎng)上咨詢等）與隊(duì)外的任何人（包括指導(dǎo)教師）研究、討論與賽題有關(guān)的問題。我們知道，別人的成果是違反競(jìng)賽規(guī)則的, 如果引用別人的成果或其他公開的資料（包括網(wǎng)上查到的資料），必須按照規(guī)定的參考文獻(xiàn)的表述方式在正文引用處和參考文獻(xiàn)中明確列出。我們鄭重承諾，嚴(yán)格遵守競(jìng)賽規(guī)則，以保證競(jìng)賽的公正、公平性。如有違反競(jìng)賽規(guī)則的行為，我們將受到嚴(yán)肅處理。我們?cè)试S數(shù)學(xué)中國(guó)網(wǎng)站()公布論文，以供網(wǎng)友之間學(xué)習(xí)交流，數(shù)學(xué)

2、中國(guó)網(wǎng)站以非商業(yè)目的的論文交流不需要提前取得我們的同意。我們的參賽隊(duì)號(hào)為：2326我們選擇的題目是：Problem A(MCM): The slimline seats on the airplane參賽隊(duì)員(簽名) ：隊(duì)員 1：周琛隊(duì)員 2：文楷隊(duì)員 3：范川參賽隊(duì)教練員 (簽名)： 楊志霞更多數(shù)學(xué)建模資料請(qǐng)關(guān)注微店店鋪“數(shù)學(xué)建模學(xué)習(xí)交流” /RHO6PSpA第四屆“認(rèn)證杯”數(shù)學(xué)中國(guó)數(shù)學(xué)建模國(guó)際賽編 號(hào) 專 用 頁(yè)參賽隊(duì)伍的參賽隊(duì)號(hào)：2326競(jìng)賽統(tǒng)一編號(hào)（由競(jìng)賽組委會(huì)送至評(píng)委團(tuán)前編號(hào)）：競(jìng)賽評(píng)閱編號(hào)（由競(jìng)賽評(píng)委團(tuán)評(píng)閱前進(jìn)行編號(hào)）：Team#2326Pa

3、ge 3 of 30The Design of Seat Back Curve Based on NonlinearProgramming and SmoothingAbstractPeople have begun to convert their mind in choosing a way of moving as livingstandard rising, and the people to go abroad have increased rapidly as the trend of economic globalization,have brought great pressu

4、re to airlines to increase their capacity to meet the transport demand, so the slimline seats on the airplane emerge at the historic moment. It is because that the people have begun to consider making the comfort on the body and mind as the primary as nowadays, 0humanization0 should be put on a grea

5、t attention so that he airline can triumph in the 0space war0 whichever can does well in designing the most comfortable seat back even though it has a thinner backplate and less padding.So how to design a seat back curve to increase the comfort of slimline seats is a problem need to be solved urgent

6、ly.The interpersonal spines are different in the modalities, it is related to so many factors, in this thesis we will focus on the design method of seat back curve, not collect a large amount of data to process. Through the analysis of studies, we found that the curve fitted by the least squares Gau

7、ss curve fitting method is in conformity with the physiological curve of spine, so we bring out the Gauss curve fitting model. And then, introduces the abstraction of body posture in resting, carries on the theoretical analysis combined with biological engineering and receives five stress points and

8、 analyses them. According to the information, we gives a nonlinear optimization model and uses Lingo14.0 to calculate, obtains the location information of the stress points, using Gauss curve fitting model for curve fitting, and carries on the smooth optimization to ensure the five points are the so

9、le extremum points of the curve corresponding, this curve is been considered as a man0 s spinal curve in the condition of nature, at the same time, the joint curve is the best seat back curve to let a person feel more comfortable.Due to the method of the stress point in the nonlinear optimization mo

10、del system is sketchy, this thesis considers carrying on a further optimization, so we bring out a new optimization model based on body pressure distribution. In this model, we define 8 indexes to discuss. If the condition permitted we can obtain a lot of data caused by pressure change on different

11、posture(within a certain range) dynamically by theuse of the pressure sensor technology, make statistics and analysis, and then find aTeam#2326Page 4 of 30pressure range in which comfort level can achieve a higher point. After that, makingsome modification on the basis of the concept of time series

12、method, get pressure data with the change of the angle between the seat back and seat of each part of spine resulting in the model, and then analyzes and predicts in combination with comfort evaluation model, find out the angle of seat backplate that comfort level can achieve a high place within the

13、 scope of the adjustable angle as well as the human body pressure distribution, at this time, the pressure distribution on human0 s spine can be got after its digital, push reversely to get the seat back curve jointed better with the modality of spine.Because of time and technical limitations, the c

14、omprehensive model this thesis pro- posed only be put forward the basic thoughts, but this model is practical which can be inferred from the literature. Simulation of artificial intelligence perhaps can be realized if this model can be blended in the idea of finite element method. Thus, it is the ta

15、skwe are likely to attempt.Key words: Set back curve ; Ergonomics ; Least-squares fitting ; nonlinear pro-gramming ; Body pressure distributionTeam#2326Page 5 of 30Contents1Introduction62The Description of Problem73Models3.1Description of Models . . . . . . . . . . . . . . . . . . . . . . . . . . .

16、.88811111111.23.1.3Terms in the Thesis . . . . . . . . . . . . . . . . . . . . . . . . .Assumption of Models . . . . . . . . . . . . . . . . . . . . . . .Symbols of Models . . . . . . . . . . . . . . . . . . . . . . . . .3.2The Foundation of Seat Back Curve Model . . . . . . . . . . . . .

17、. . ..2Gauss Curve Fitting Model. . . . . . . . . . . . . . . . . . . .Nonlinear Programming Model. . . . . . . . . . . . . . . . . .4Future Work205Advertising Material21Team#2326Page 6 of 301IntroductionThe development of the airlines has reached an unprecedented period of prosperitywith th

18、e development of society and the progress of science and technology. Along with the improvement of people0 s standard of living, more and more people are likely to have a trip by air because of its convenience and swiftness, and the price of economy class is not so high. What0 s more, the evolution

19、of economic globalization indicates that the contact among the states is becoming more and more closely, and the communication is increasing. This phenomenon shows an upward trend in the number of international flights, and the rapid growth of the number of the passengers who choose to take flight a

20、lso puts a great pressure on the aviation transportation undoubtedly. In view of the fact that increases the number of the flight constantly is unrealistic, so it can be considered to improve the capacity of airlines.In order to increase the plane0 s capacity and keep passenger0 s moveable space doe

21、snot decreased, most airlines choose to install a new type of seat for their planes, called slimline seat, to replace the old type. This kind of seat have new structure and covered by more slight fabric. This new seat is more light than the old and indeed increases the capacity of plane. However, it

22、 also rouses the complain all over the world. People feel very uncomfortable when sitting on it because of its thin backplate and little padding. As we all know, the past few months have seen several flying accidents. Though more and more people like to go out by plane, some people may hesitate whet

23、her to choose it considering the question of safety. In this case, if the seats cannot meet the comfortableness of the seat will receive more attention. As a result, the uncomfortable and thin seats will become the receiving end. Maybe passengers may wonder whether it is because the low price of Eco

24、nomy class that the airlines ignore the needs of passengers and provide to then less better services. Thus the aviation industry will get into crisis.After all, the passengers who take economy class are their major passenger sources.To sum up,the comfortableness of passengers seats is really a probl

25、em urgently to be solved. How to design a good seat that can meet the requirements of the human- machine engineering is a problem that the aviation industry is now facing.Under the condition of having related spinal data, through the analysis to find arelatively optimal curve fitting method to make

26、the curve that fitted out by this method can match the physiological curve of most human0 s pines.Team#2326Page 7 of 302The Description of ProblemAn airline seat is a seat on an airliner in which passengers are accommodated forthe duration of the journey. Some airlines are now introducing new 0sliml

27、ine0 seats in economy class. These seats,in addition to weighing less, theoretically allow airlines to increase capacity without significantly affecting passenger comfort. These seats may or may not feature moveable headrests, and generally do not feature adjustable lumbar support. Slimline seats ar

28、e being further refined, liberating more passenger space. The common point of them is a thinner backplate and less padding. However, many passengers have expressed displeasure with these seats.Task 1: Without changing the structure of the premise, how to design seat back curve, in order to make the

29、seat more comfortable?In order to solve this problem, we can not get a lot of data of different human spine because of the limited time. Therefore, this paper consider focusing on method of put forward.What0s more, we find the required data by writing C program which can makethe human body vertebra

30、image digitization and coordinated. We randomly pick some points from human spine picture to make a scatter dia- gram. Next we choose some different curve to fit the scatter diagram. So we get some approximate human spine curve in natural condition. Then we compare these curves with human spine imag

31、e to pick a better curve preliminarily. By abstracting analysis on the human body vertebra structures, we have found points respectively on the force analysis, based on the analysis of the existed literature to find the values of parameters which are required and the scope of variables, and nonlinea

32、r optimization model is established to find points, then find the coordinates of the stress point according to make them coordinate. Now we can take advantage of the former step in the selection of the optimal function fitting types on the fitting to get the seat backrest curve which we expect.Task

33、2: How to optimize the seat backplate curve and padding, in order to make the seat more comfortable, without changing the main internal structure?To solve this problem preferably, we further optimize the model above. In this present model, we just calculate the approximate index of force bearing poi

34、nts and do the rough force analysis, so the error will be dissatisfactory. To minimize the error, we use pressure sensing technology to get seats pressure distributions of different humansitting posture. And then, we code program to analyze these pressure data. Next,Team#2326Page 8 of 30using these

35、data to optimize our curve model. At the end, we build a neural networkto evaluate our curve model.figure 1.The overall flow chart to solve the problem, as shown inFigure 1: Generate the backrest curve diagramIn additionto the above-described problems, this thesis will give an advertisingmaterial fo

36、r the airline to describe our design features and advantages concisely in thesame time.3Models3.1Description of Models3.1.1Terms in the ThesisHuman spine : Human spine is composed of 33 vertebrae (12 pieces of cervicalvertebra, 7 pieces of thoracic vertebra, 5 pieces of lumbar vertebra , and sacrum

37、and coccyx which are totally 9 pieces) that are combined with joint, ligaments and inter-Team#2326Page 9 of 30vertebral disc. Spine is upper supported by skull, bottom supported by hip, in thecentral attached by ribs, and as the back wall of thorax and abdominal and pelvic cavity. Spine has the func

38、tion of supporting the trunk, protecting viscera, protecting cord and conducting movement. As shown in figure 2.Figure 2: The human body vertebraErgonomics : So-called ergonomics, which is the research method of such subjectsas anthropometry, human body mechanics, physiology of labor, labor psycholo

39、gy, stud- ies on the human body structure characteristic and the function characteristic, provides such human body structure characteristic parameters as the size, weight, surface area, specific gravity, center of gravity of each part of human body, and the relationship and moving range of each part

40、 of the human body in the activity . Also it provides such characteristic parameters as the range of output, and the habits on the movement of each part of human body the function of the body, analysis functional characteristics of sensory organs , such as human visual, hearing, touch, skin sensatio

41、n; It analysis a variety of physiological changes, energy consumption, fatigue mechanism and people0 s ability to adapt to all kinds of labor load when people are at work. It discusses the factors that affect mental state at work and the influence of psychological factors on the efficiency and so on

42、.Artificial Neural Networks : Artificial Neural network (abbreviation for ANNs) which is called short for Neural Networks (NNs) or Connection Model, it is a kind of algorithm mathematical Model of imitation on animal Neural network behavior char- acteristics, do information processing with distribut

43、ed parallel. The network dependson the complexity of the system, by adjusting the internal relations between a largeTeam#2326Page 10 of 30number of nodes connected, so as to achieve the purpose of information processing.The working principle of the abstraction is shown as figure 3.Figure 3: The Work

44、ing Principle of Artificial Neural NetworksSensing Technology : To obtain information on various types of sensors, theyhave a variety of physical, chemical or biomass of sensors. According to the convex theorem of information theory, the function of the sensor and quality determines the amount of in

45、formation and information quality of natural information acquired from information sensing system, is the first key to the structure of high quality sensing technology system. Information processing includes signal preprocessing, post pro- cessing, feature extraction and selection, etc. Identificati

46、on and classification is the main task of the processed information for Recognition. It uses the relationship model between recognized(or) diagnosed objects and the characteristic information and do identification, comparison, classification, and judgment to input feature information.Least Squares :

47、Least squares method (also known as the least square method) is a kind of mathematical optimization techniques. Its data by minimizing the error sum of squares to find the best match function. Can be easily obtained by means of least square method of the unknown data, and make the the sum of the squ

48、ares of the error between actual data and calculated data to a minimum. The least square method can also be used for curve fitting. Other optimization problems can also be used by minimizing or maximizing entropy energy use least square method to express.Curve fitting :Curve fitting refers to choose

49、 the appropriate type of curve to fittingthe observed data, and the fitting curve equation analysis of the relationship between the two variables.Team#2326Page 11 of 303.1.2Assumption of Models Assuming that the rigidity of the seat frame can meets the requirements; Regardless of the seat cushion, s

50、eat back curve is only considered comfort brought by the change; Don0 t consider the cost of materials.3.1.3Symbols of ModelsSymbolDescriptionH LTilt angle of seat backBackrest heightThe distance between two seats3.2The Foundationof Seat Back Curve ModelAs a matter of time, we can not get the data o

51、f human spine about different agestages, sexes , nationalities and races under natural state. Because airplane seats are designed for most of the normal human body, this paper extracts data points from ordinary normal spinal image of the body. In the first place, we use C+ programming. According to

52、a series of transformations based on the differences of the spine image grey value, we achieve the extraction of data points. Then, by making these data points coordinated and digitization under coordinate system, we can get a point column from the curve of the human body vertebra. These points are

53、used in the model we buildthe next as the known data points.3.2.1Gauss Curve Fitting ModelGiven these data of points : xi, yi, here yi = fxi(i = 0, ., m), to find a functiony = S(x) with the given data (xi, yi), i = 0, 1, ., m fitting, If define the error is i = S(xi) yi(i = 0, 1, ., m), = (0, 1, .,

54、 m)T , Set 0(x), 1(x), ., n(x) is the linearly independent functions on Ca, b,find a function S(x) in =span0(x), 1(x), ., n(x) to minimize the error sum of squares.XmXmXm2222kk =(x ) =(x )S (x ) y = min(x )S(x y )(1)iiiiiii2iS(x)i=0i=0i=0Team#2326Page 12 of 30In the above type (x) 0 is the weight fu

55、nction on a, b ,it stands the differencesin the proportion of different data (xi, f (xi) ,and here ,S(x) = a00 + a11 + . + ann (n m)(2)Using least square method for fitting curve of the problem, is in theform such as(2) in the S(x) for a function y = S(x), to gain the minimum of (1). It is transform

56、edinto the problem of finding out the minimum point (a0, a1, ., an) of the function with many variables.XmXn2(x ) f (x )I(a , a , ., a ) =(x )a (3)01nij jiii=0j=0By the necessary conditions of multivariate function extreme value, there ismnXXI= 2(x )a (x ) f (x ) (x ) = 0, k = 0, 1, ., n(4)ij jiikiaki=0j=0Defines thatXm( , ) =(x ) (x ) (x ),jkijikii=0mX(f, ) =(x )f (x ) (x ) d , k = 0, 1, ., n,kiikiki=0So(k, j)aj = dk, k = 0, 1, ., n.(5)The type can be written as a matrix formGa = d,(6)Herea = (a0, a1, ., an)T , d = (d0, d1,