I. Asking questions

Steps and stairs are very common in our daily life, and we have to walk every day. Good step design can not only save the time of going upstairs, but also minimize physical consumption. But unreasonable design will make people go upstairs time-consuming and laborious, and even dangerous. So we can't help but ask, how to design the length-width ratio of the steps to achieve the best? (The following mainly discusses the process of going upstairs, and finally involves the discussion of going downstairs. )

As the first step to solve the problem, we will first prove the existence of this optimal design. The following two pictures show two different types of steps.

When the total height, step width and physical consumption remain unchanged, and the height h of the steps is small enough, the number of steps will be large enough, and the time t for going upstairs will tend to infinity. So we won't go upstairs. If H is made large enough, and people's leg movement ability is limited, then the increase of work in each step will inevitably lead to the accumulation and growth of climbing time, which is also unacceptable to us. Due to the continuous changes of various states, we can draw a conclusion that there is such an H that T is the minimum. Similarly, when the step length r is small, people can't stand still. When r is large enough, time t tends to infinity. So we have every reason to believe that the optimal R and H exist. The analysis here only depends on perceptual knowledge and geometric intuition. Below, we will give a reasonable answer as far as possible from a mathematical point of view.

Two. problem analysis

Symbolic representation:

M human body mass

acceleration of gravity

L human calf length

Normal walking speed of V people

Leg strength when going upstairs

Total height of stairs

H step height

R step size

P the most comfortable output power when human body climbs stairs at height h.

C people have long feet.

To analyze this problem carefully and comprehensively, we can decompose the whole process of people going upstairs and set each step of going upstairs as a unit, then we can draw a sketch of the human movement process. Considering that going upstairs is a very complicated human dynamic process, some artificial assumptions will be necessary in order to grasp the main contradiction and simplify the problem.

The model assumes: 1, and the front end of the foot touches point B every step.

2, a person's total weight can be regarded as a particle and concentrated on O (equivalent to concentrated on N), and other parts have no weight.

3. Walk the same distance (step width) every step and keep moving forward.

4. The rising power of the human body comes from the force F supporting the legs, which is related to H. The magnitude of F is constant, and it always keeps the ON direction when H is fixed.

5. The work done in the process of going upstairs is only in the DN section, and people always go upstairs with the so-called most comfortable feeling (P unchanged).

6. The step length is greater than or equal to feet.

Motion decomposition: climbing stairs can be regarded as two sports processes.

1. (from m to n) If people want to climb the steps, leaning forward with their center of gravity will be the first step. After all, people are moving forward. In order to exert force on point D, it is reasonable to move point M to point N ... and this process is very close to the state when people walk on the flat ground (here, the speed is equal, and the direction of V is approximately horizontal). In order to simplify the calculation, the work done in this section can be made negligible (because our main contradiction is going upstairs, and the change of work done in this section is equivalent to the difference between walking 5 meters and 10 meters on the flat ground, which is very small in the eyes of normal people)

2.(N points stand vertically and return to the initial state) The work done in this process is the contribution of F (here, the leg buckling is very similar to the release process in the shot put model in class, because the main contradiction at that time is the initial velocity of the ball, so it can be regarded as a linear relationship approximately, but the focus at this time is on this buckling process, so it is assumed that nature is different from the model mechanism). Then, according to the knowledge learned in biology class, we can know that the movement of human legs depends on the expansion and contraction of muscle cells (the expansion and contraction direction can only be along the leg direction), so all muscles can be regarded as a force, and the direction is always along the leg direction, with the same size (in fact, f changes with the angle, which can be set as a constant to simplify the problem). Considering that the work that people do when they ascend in two steps is actually done by non-conservative forces (not w=mgh), it is very simple and intuitive that we must do work in 10 and step 2 when climbing the same height of 2 meters. The root cause of this difference lies in the angle between the bearing capacity of the leg and the direction of stress (that is, the higher the steps, the more extra work we do). Therefore, it is necessary to measure the concept of "leg work" from a mathematical point of view, and hypothesis 4 will be necessary. Secondly, we should measure the two concepts of "comfort" and "fatigue". Usually, the main reason for our short-distance fatigue is that the exercise intensity of the legs is too high, that is, the power P is too high. This makes it possible to measure "comfort".

Three. Get data

Calculation of walking speed V: First of all, the so-called "normal speed" is a vague concept, but it exists objectively. In order to get the normal walking speed of people as much as possible, and the error is as small as possible, the method of multiple measurements is adopted here. You need to do the experiment yourself. The floor near the entrance of the house is paved with square bricks, each brick is square with a side length of 0.48 meters. This provides convenience for distance measurement. Walking at the normal speed with maximum self-control ability, it is stipulated that when five bricks are passed, the time will be recorded, and the point will be set to zero (in order to remove the acceleration section). Finally, 1 1 group data is obtained.

Distance (meters) Time (seconds)

1 2.4 2.03

2 2.88 2.42

3 3.36 2.78

4 3.84 3.22

5 4.32 3.57

6 4.8 3.97

7 5.28 4.47

8 5.76 4.8 1

9 6.24 5. 19

10 6.72 5.53

1 1 7.2 6.05

The following figure is obtained by fitting in matlab. The linear polynomial is y = 0.012909+0.83186x, so the normal walking speed is1.202m/s.

Weight of 53 kg, leg length of 0.47 m and foot length of 0.26 m can be measured accurately. Only the power p is unknown, but because we assume its size is constant, we can inverse the solution according to the relationship in the subsequent model solution.

Four. Establishment of model

Assume that the total number of steps is (if there is a score, it can be approximately regarded as the number of times taken in each short time). This error can be ignored)

Let the time of process 1 be, and the total time of substituting the relation into available process 1 be

The total time of process 2 is

Where the functions are H, L, F and P, because we assume that points M and N have roughly the same height. Then it is a function that has nothing to do with X. If the total time

Minimum value, x must be the minimum value. So it is available. We come to the conclusion that the width of the steps should be designed to be close to the width of the feet. From this, the following figure A is obtained, and the change of H is discussed on this basis.

Because we first assume that the size of f is constant. If F can drive the human body to move upwards, Fy must be at least equal to mg, so we take it with the least effort. At this point, we have decomposed F. Therefore, in the process of moving from point N to point S, the work required by F only needs to be calculated separately for Fx Fy.

We carefully analyzed the movement process and enlarged it into a B diagram.

Working in Fy direction when the step height is h: let the length of NNm be a variable m, and when the Nm moves from n to s. M, it changes from 0 to h.

Through differential analysis, when m changes △m,

Where S(△m) is the vertical moving distance of Om.

Integral of m

2. When the step height is h, work in Fx direction:

Differential element analysis, plus △m, we get

Divide △m by both sides to make △m→0. therefore

Where S(m) is the length of PmOm. Integral of m

Because we assume that f is a function of h (h takes a constant timing). So take it.

All in all, the total time we spent upstairs.

Let's determine the minimum value of t from this formula and put the parameters

P to be determined.

All the above calculations can be done with maple. The calculation process is as follows

t:= m-& gt； sqrt(0.47^2-((2*0.47-h+m)/2)^2)；

diff(t(m)，m)；

e:= m-& gt； -sqrt(0.47^2-((2*0.47-h+m)/2)^2)* 1/2/(.2209-(.4700000000- 1/2*h+ 1/2*m)^2)^( 1/2)*(-.4700000000+ 1/2*h- 1/2*m)/0.47；

int(e(m)，m=0..h)；

wy:= h-& gt； (2*0.47*h-h^2/2)/(4*0.47)；

f:= h-& gt； (2 * 0.47 * 53 * 9.8)/(2 * 0.47-h)；

wx:= h-& gt； & gt.4999999999*h-.2659574468*h^2

From this, we find that WX and WY do basically the same work. So finally, the total time is expressed as

& gtf:= h-& gt； h*( 1.2*(2*0.47*53*9.8)/(2*0.47-h)*(.4999999999*h-.2659574468*h^2+.5*h-.2659574468*h^2)+0.26*p)/(h*p* 1.2)；

Moreover, according to the above results, we can observe the relationship between the work done by human legs (Wx(h)+Wy(h)) and the actual effective work Mgh.

Process diagram of system changing with h.

The red line is the total work done by human legs, and the yellow line is the effective work Mgh. This change is also in line with our feelings. For example, with the increase of H, we will feel more and more laborious to step on the steps. The greater H, the more obvious this change is.

Then several groups of experiments are carried out to determine the approximate value of P, choose different stairs, walk from bottom to top at normal speed (without feeling tired), and record the elapsed time. According to the hypothesis and the above formula, p is obtained respectively, and the following table is obtained.

Step number n step height h total height h time t power p

1 20 0. 17 3.4 18. 1 1 142.34

2 18 0. 15 2.7 14.83 140.49

3 25 0. 14 3.5 18.92 133.09

4 16 0. 18 2.88 15.06 144.3 1

5 20 0. 16 3.2 16.87 146. 18

6 22 0. 17 3.74 18.87 152.94

7 20 0. 15 3 15.79 148.92

8 18 0. 16 2.88 14.9 1 149.79

9 16 0. 17 2.72 15. 10 134.85

Practice has proved that P has little change with the total height h and h, which shows that our previous assumptions are basically reasonable. Here, the average of 9 measurements is taken as p, so we get P= 143.66.

We analyze t in the first case. Take H=3.4.

& gtf:= h-& gt； 3.4*( 1.2*(2*0.47*53*9.8)/(2*0.47-h)*(.4999999999*h-.2659574468*h^2+.5*h-.2659574468*h^2)+0.26* 143.65)/(h* 143.65* 1.2)；

plot(f(h)，h=0. 1..0.5);

From the image, we observe that there is such an H, and the total time is the least. That is to say, given a certain H's upstairs time, we can calculate the ideal height of H's minimum time on this power P. In the above figure, the time to reduce from 0. 19 to 0.24 meters is about 0.2 seconds, and the optimization of this time is so small (0.2 seconds) that it can be ignored (it can be approximated) The period of rapid time reduction is from 0. 1 to 0. 19. So in order to make the leg strength as small as possible, we might as well set h to 0. 19 meters.

So we have to ask, how reliable is this model? Because v P is roughly calculated, we will analyze the sensitivity of these two parameters.

Plot3d(f(h, v), h=0. 1..0.5, v =1.1..1.3, axis = boxed);

Plot3d(f(h, p), h=0. 1..0.5, p= 140.. 154, axis = boxed);

It can be seen from the three-dimensional graphics that the model is reliable. The sensitivity analysis method used by the teacher in class is not used here, because I just want to visually show the continuous dependence of the solution on the parameters. It seems not intuitive to use only discrete data.

So far, after calculation, my best walking height should be about 0. 19 meters, that is to say, this height can fully and effectively use my normal strength, so that the total time to go upstairs is the shortest, and I will not feel tired if I exceed the limit.

By the way, here is an explanation of the process of going downstairs. In a short distance, the process of going downstairs can be approximately regarded as a process in which the legs do zero work and the work is completely completed by gravity. Because gravity is a conservative force, the time to go downstairs should have nothing to do with H approximation. But why does going downstairs for a long time make us feel tired? The reason may be the buffering force when going downstairs. After all, people are different from wooden blocks and small balls, and falling too fast will cause discomfort to the legs and body, so the legs always have to do some work to make them fall slowly and smoothly.

I introduce the variable of buffer time here, where t is the actual total time to go downstairs, L is the step width, and V is the horizontal walking speed. Obviously, it is the sum of buffering (delay) time. For most normal people, in the process of going downstairs for a short distance, within the normal range of H (the range calculated above), it can be approximately regarded as 0. Then we can only discuss the process of going upstairs. However, can it be ignored forever? The answer is obviously no, for example, when H is big, it is a function of H and H (the influence of H cannot be ignored), and some special groups such as the elderly and the disabled will be quite big, so the process of going downstairs should be considered separately.

A test of verb (verb's abbreviation) model.

Due to the particularity of the above data, the model is too specialized. After all, I am not walking alone. However, I am a normal person, even considering the uncertain factors of many people's parameters, the change will not be too great.

According to the survey, all the steps in the campus range from 0. 16 to 0.2m, from the lowest step in front of the science and technology building to the highest step in front of the fourth canteen. The width is similar to the length of the foot, which shows that the conclusion of the model is barely acceptable (although not so accurate). This is equivalent to testing the model to a certain extent (because the height of the steps can be adjusted appropriately according to the practice, and the inappropriate height must not exist, or it will be modified or improved in the next construction).

Further reference can be made to the relevant provisions of the Code for Architectural Design GB 50096-1999, which came into effect on June 1 999: "The stair tread width should not be less than 0.26m, the tread height should not be greater than 0. 175m, and the slope is 33.94, which is close to the comfort standard. Among them, 0.26 must be the foot length, and 0. 175 is the best height. (This result may be the result of relevant mechanics and statisticians, and it should be more authoritative data. )

Error analysis: Through the above inspection, it can be seen that the calculated results are indeed different from the actual ones, and the calculated H is too large. The reasons for this deviation are as follows.

(1) People's weight difference

(2) Difference in height and leg length

(3) the difference of human foot length

(4) Forward leaning speed (here, it is taken as walking speed, but the first process is only forward leaning process, and its speed must be greater than walking speed, so it is not easy to measure, so the error must be inevitable).

(5) The function of f changing with leg movement is not accurately known (it will involve complex human dynamics, and due to limited knowledge, it has to be assumed to be constant in order to simplify the complexity. There is not much deviation in the calculation results, which shows that the assumption is basically reasonable, but the error is inevitable)

(6) The difference of people's normal strength, such as the amount of exercise that professional athletes and ordinary people can bear must be different.

Therefore, if the above data can be accurately known, there is reason to believe that the error of the calculation results will be small. The model will be more reliable.

Importance of intransitive verb model

Through the analysis of this model, the approximate relationship between F v P c L M is found. However, this also raises a question, is the provision in the code for architectural design GB50096- 1999 too one-sided? The data of 0. 175m must be the statistical average. In some specific situations, it must be further stipulated, for example, the height of steps in middle school buildings and university buildings can be equal. However, the steps of kindergartens, nursing homes and rehabilitation centers must be stipulated separately. Otherwise, due to the improper height of the steps, danger will occur. If the relevant data are obtained, the optimal height can be calculated according to the model, thus expanding the content of architectural design specifications.

end