^{1}

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^{2}

^{2}

^{2}

^{1}

^{2}

Air brake system is one of the common braking methods for buses and trucks; its excellent performance guarantees the safety of the vehicle and the stability of the braking. As an important part of the pneumatic brake system, the brake pipe is an important factor influencing the pressure response time of the pneumatic brake system. Based on the exploratory experiment of pneumatic brake pipe, the influence of pipe length, pipe diameter, inlet sonic conductance, initial pressure, and supply pressure on pipe pressure response time was analyzed by fuzzy gray correlation analysis method. The results show that tube length is the most important factor affecting the pressure response time. Combined with the analysis results of gray correlation degree, the experimental scheme of the response time of the pneumatic brake pipe was designed by the response surface experimental design method. Based on the multiparameter analysis method, the influence of the experimental parameters on the pipe pressure response time was analyzed. Based on the experimental data, the form of calculation formula is derived by dimension analysis method, which provides a theoretical basis for the selection of pneumatic brake pipes and the design of air brake system.

As one of the most commonly used braking methods, pneumatic braking has been widely used in buses and trucks [

According to the arrangement and function of the pneumatic brake pipe in the pneumatic brake system, the pneumatic brake pipes are mainly divided into three types, as shown in Table

Types of pneumatic pipe.

Pipe types | Features | Positions |
---|---|---|

Energy supply pipe | Supply compressed air | Between the air compressor and the air tank |

Control line | Control air circuit opening and closing | Between the air tank and the relay valve |

Actuation line | Actuator | Between the pedal valve and the relay valve |

Table

Schematic diagram of one-quarter brake circuit.

For pneumatic pipelines, many scholars study the calculation method of pipelines. Cai [

According to these descriptions above, the analysis of the pressure response time of pneumatic brake pipe mainly has some deficiencies as follows. (1) The experimental design method in the international standard only gives the design scheme of the general pneumatic circuit, which cannot provide theoretical support for the selection of experimental parameters, resulting in lack of basis for the selection of parameters in the experimental design. (2) The research on the pressure response time of pneumatic brake pipes mainly focuses on the flow characteristics of pneumatic brake pipes, which is insufficient for guiding the design of pneumatic brake systems. Therefore, there is a lack of a formula for calculating the pressure response time of a pneumatic brake pipe with appropriate accuracy and convenient calculation.

In this paper, the pneumatic brake pipe is taken as the research object, and the pressure response time of the pneumatic brake pipe is studied based on the experiment. The parameters affecting the pressure response time of the pneumatic brake pipe are analyzed. Combined with the requirements of the pneumatic brake system for the pressure response time of the pneumatic brake pipe, the calculation formula of the pressure response time is derived, which provides a theoretical reference for the design of the pneumatic brake pipe.

In order to design the experimental scientifically, it is necessary to clarify the parameters affecting the pressure response time of the pneumatic brake pipe and its laws. Grey correlation degree [

Exploratory experimental schematic for pressure response time.

Figure

Experimental curve of brake pipe pressure response.

In this paper, the experimental data of brake pipe pressure response time is obtained based on the exploratory experiment of pneumatic brake pipe. The influence law of influence parameters on pressure response time is analyzed by fuzzy gray correlation degree. The key influence parameters of pressure response time of pneumatic brake pipe are determined, which provides a theoretical reference for the selection of the parameters of the pipe’s pressure response experiment.

Here,

Considering the influencing factors involved in the air brake system, the tube length

The matrix of gray correlation degree reflecting the behavior characteristics of the system is a reference sequence and is as follows:

Here,

The reference sequence was obtained by experiments from the parameters of the comparison sequence. According to the engineering application, the range of experimental parameters is set as shown in Table

Exploratory experiment parameter range setting.

Parameter type | Parameter range |
---|---|

| 8-12 |

| 5-20 |

^{3}/(s·bar) | 1-4 |

_{0}/MPa | 0.1-0.3 |

| 0.4-0.65 |

Experimental data for comparing sequence and reference sequence.

No. | | | ^{3}/(s·bar) | _{0}/MPa | | |
---|---|---|---|---|---|---|

1 | 12 | 20 | 4 | 0.1 | 0.4 | 386 |

2 | 8 | 12.5 | 2.5 | 0.2 | 0.525 | 170 |

3 | 12 | 5 | 4 | 0.3 | 0.65 | 59 |

4 | 10 | 12.5 | 2.5 | 0.2 | 0.65 | 229 |

5 | 10 | 12.5 | 2.5 | 0.3 | 0.525 | 170 |

6 | 12 | 20 | 4 | 0.3 | 0.65 | 301 |

7 | 8 | 5 | 1 | 0.3 | 0.65 | 110 |

8 | 12 | 20 | 4 | 0.3 | 0.4 | 196 |

9 | 8 | 20 | 4 | 0.1 | 0.4 | 345 |

10 | 8 | 20 | 4 | 0.3 | 0.4 | 181 |

Here,

Table

Correlation coefficient of each parameter.

| | | _{0} | |
---|---|---|---|---|

0.74 | 0.61 | 0.69 | 0.50 | 0.56 |

Here, r_{2} indicates membership;

Membership of each parameter.

Coefficients | | | | _{0} | |
---|---|---|---|---|---|

r_{2} | 0.96 | 0.90 | 0.92 | 0.77 | 0.85 |

Here, r is fuzzy relevance. The fuzzy relevance of each parameter is shown in Table

Fuzzy correlation degree of each parameter.

Coefficients | | | | _{0} | |
---|---|---|---|---|---|

r | 0.85 | 0.76 | 0.80 | 0.63 | 0.71 |

It can be seen from Table

Therefore, pipe length, pipe diameter, inlet sonic conductance, and gas supply pressure cause the fuzzy correlation degree to exceed 0.7, which are parameters with obvious influence degree. Compared with other parameters, the initial pressure has no obvious influence on the pressure response time.

Based on the fuzzy gray correlation analysis, the experimental design of the pneumatic brake pipe pressure response time is conducted by using the CCD design of experimental method [

According to QC/T 35-2011 [

Test bench for line pressure response time.

GB 12676-2014 “Technical requirements and test methods for brake systems for commercial vehicles and trailers” stipulates that when the pressure measured from the start of the brake pedal to the control line joint reaches 10% of the stable value, pressure response time of pneumatic brake pipe shall not exceed 0.2 s; when it reaches 75% of the steady state value, pressure response time of pneumatic brake pipe shall not exceed 0.4 s.

_{1}.

_{2}.

The reading formula for the pressure response time of the inflation process is

The average of the three measured pressure response times is taken as the final pressure response time, which helps to reduce system errors.

Multiparameters experimental data.

No. | | | ^{3}/(s·bar) | | |
---|---|---|---|---|---|

1 | 8 | 5 | 3.4 | 0.5 | 75 |

2 | 12 | 12.5 | 2.2 | 0.4 | 187 |

3 | 12 | 5 | 3.4 | 0.3 | 77 |

4 | 8 | 20 | 3.4 | 0.5 | 333 |

5 | 10 | 12.5 | 2.2 | 0.4 | 155 |

6 | 8 | 20 | 3.4 | 0.3 | 319 |

7 | 8 | 5 | 1 | 0.5 | 111 |

8 | 10 | 12.5 | 2.2 | 0.4 | 155 |

9 | 10 | 12.5 | 1 | 0.4 | 281 |

10 | 12 | 5 | 1 | 0.3 | 170 |

11 | 12 | 5 | 1 | 0.5 | 181 |

12 | 12 | 5 | 3.4 | 0.5 | 81 |

13 | 8 | 12.5 | 2.2 | 0.4 | 184 |

14 | 10 | 12.5 | 2.2 | 0.3 | 151 |

15 | 12 | 20 | 3.4 | 0.3 | 332 |

16 | 8 | 20 | 1 | 0.5 | 480 |

17 | 8 | 20 | 1 | 0.3 | 452 |

18 | 10 | 12.5 | 3.4 | 0.4 | 135 |

19 | 8 | 5 | 1 | 0.3 | 106 |

20 | 8 | 5 | 3.4 | 0.3 | 74 |

21 | 12 | 20 | 1 | 0.5 | 824 |

22 | 12 | 20 | 1 | 0.3 | 786 |

23 | 10 | 5 | 2.2 | 0.4 | 83 |

24 | 10 | 12.5 | 2.2 | 0.5 | 157 |

25 | 10 | 20 | 2.2 | 0.4 | 356 |

26 | 12 | 20 | 3.4 | 0.5 | 351 |

Reading of pressure response time.

In order to further confirm the influence of each parameter on the pipe pressure response time, a multiparameter analysis method was used to clarify the interaction between the parameters. Figure

Effect of multiparameter changes on inflation time: (a) pipe diameter and length; (b) pipe diameter and inlet conditions; (c) pipe diameter and supply pressure; (d) pipe length and entry conditions; (e) tube length and gas supply pressure; (f) entrance conditions and supply pressure.

As shown in Figure ^{3}/(s·bar), the pressure response time when pipe diameter is 8 mm is greater than that when pipe diameter is 9 mm. The pipe pressure response time decreases first and then increases with the increase of pipe diameter. When the sonic conductance is 3.4dm^{3}/(s·bar), as the pipe diameter increases, the pressure response time decreases, so the sonic conductance and the pipe diameter are coupled. As the pipe diameter increases, the volume acts more than the flow rate, and the pipe pressure response time increases. Moreover, when the sonic conductance is large enough, the pipe diameter limits the flow rate. As the pipe diameter increases, the flow velocity gradually increases, and the sonic conductance dominates. Therefore, as the sonic conductance increases, the pipe pressure response time decreases.

The influence of pipe length, pipe diameter, and gas supply pressure on pipe pressure response time is unidirectional. The pipe pressure response time increases with the increase of pipe length, pipe diameter, and supply pressure; the influence of sonic conductance on the pipe pressure response time is negatively correlated; that is, as the sonic flow conductance increases, the pipe pressure response time decreases; there is a coupling effect between the pipe diameter and the inlet sonic conductance; when the pipe diameter or sonic conductance is in the leading role, the pipe pressure response time is only affected by the dominant parameters.

To facilitate the calculation of the pipe pressure response time, the dimension analysis method is used to derive the formula for calculating the pressure response time.

Eckersten [

The formula is as follows:

Here, L indicates the length of the tube; d indicates the diameter of the pipe; Cg represents the sonic flow conductance of the pipe.

The calculation results of each experimental group are shown in Table

Comparison of sonic flow conductance values.

No. | | | ^{3}/(s·bar) | ^{3}/(s·bar) | |
---|---|---|---|---|---|

1 | 8 | 5 | 3.4 | 3.64 | 0.24 |

2 | 12 | 12.5 | 2.2 | 6.91 | 4.71 |

3 | 12 | 5 | 3.4 | 9.93 | 6.53 |

4 | 8 | 20 | 3.4 | 1.97 | -1.43 |

5 | 10 | 12.5 | 2.2 | 4.34 | 2.14 |

6 | 8 | 20 | 3.4 | 1.97 | -1.43 |

7 | 8 | 5 | 1.0 | 3.64 | 2.64 |

8 | 10 | 12.5 | 2.2 | 4.34 | 2.14 |

9 | 10 | 12.5 | 1.0 | 4.34 | 3.34 |

10 | 12 | 5 | 1.0 | 9.93 | 8.93 |

11 | 12 | 5 | 1.0 | 9.93 | 8.93 |

12 | 12 | 5 | 3.4 | 9.93 | 6.53 |

13 | 8 | 12.5 | 2.2 | 2.45 | 0.25 |

14 | 10 | 12.5 | 2.2 | 4.34 | 2.14 |

15 | 12 | 20 | 3.4 | 5.61 | 2.21 |

16 | 8 | 20 | 1.0 | 1.97 | 0.97 |

17 | 8 | 20 | 1.0 | 1.97 | 0.97 |

18 | 10 | 12.5 | 3.4 | 4.34 | 0.94 |

19 | 8 | 5 | 1.0 | 3.64 | 2.64 |

20 | 8 | 5 | 3.4 | 3.64 | 0.24 |

21 | 12 | 20 | 1.0 | 5.61 | 4.61 |

22 | 12 | 20 | 1.0 | 5.61 | 4.61 |

23 | 10 | 5 | 2.2 | 6.34 | 4.14 |

24 | 10 | 12.5 | 2.2 | 4.34 | 2.14 |

25 | 10 | 20 | 2.2 | 3.51 | 1.31 |

26 | 12 | 20 | 3.4 | 5.61 | 2.21 |

In Table ^{3}/(s· Pa)), and gas supply pressure p (Pa).

In order to get s, it is necessary to eliminate Pa and m. Since the unit of d and L is m, the formula form on the molecule becomes

In order to derive the unit s, it is necessary to eliminate the unit m of the denominator and the unit Pa of the numerator. The unit of L on the molecule is m. In order to eliminate the denominator unit m, the formula form of the molecule becomes

When

Supplementary experimental data.

No. | d/mm | L/m | Cr/dm^{3}/(s·bar) | Cg/dm^{3}/(s·bar) | p/MPa | | |
---|---|---|---|---|---|---|---|

1 | 8 | 12.5 | 3.4 | 2.45 | 0.2 | 164 | 250 |

2 | 8 | 12.5 | 3.4 | 2.45 | 0.3 | 171 | 283 |

3 | 8 | 12.5 | 3.4 | 2.45 | 0.4 | 175 | 306 |

4 | 8 | 12.5 | 3.4 | 2.45 | 0.5 | 176 | 320 |

5 | 8 | 20 | 2.2 | 1.97 | 0.2 | 315 | 558 |

6 | 8 | 20 | 2.2 | 1.97 | 0.3 | 332 | 627 |

7 | 8 | 20 | 2.2 | 1.97 | 0.4 | 342 | 677 |

8 | 8 | 20 | 2.2 | 1.97 | 0.5 | 349 | 714 |

9 | 8 | 20 | 3.4 | 1.97 | 0.2 | 309 | 490 |

10 | 8 | 20 | 3.4 | 1.97 | 0.3 | 319 | 554 |

11 | 8 | 20 | 3.4 | 1.97 | 0.4 | 327 | 598 |

12 | 8 | 20 | 3.4 | 1.97 | 0.5 | 333 | 631 |

Based on the above pressure response time calculation formula, the coefficients of the formula are fitted by 1stopt® software. The coefficients of the pressure response time formula are fitted and the optimal calculation coefficient is determined by selecting an appropriate algorithm.

Calculation formula (Cg≥Cr).

Application ranges | Formula forms | ||||
---|---|---|---|---|---|

| | ||||

| |||||

Units | Correlation coefficient | Decision coefficient | |||

| |||||

| | | | ||

m | m | m^{3}/(s·Pa) | Pa | 0.97 | 0.94 |

From Table

Figure

Calculation error of the formula.

Only when the calculation formula error is less than 5%, the calculation formula has engineering application value. When the error judgment pressure response time is ranged from 600 ms to 800 ms, the accuracy of the formula meets the requirements, in which the response time of the control line in the pneumatic brake system must not exceed 400 ms, because this calculation formula does not apply to the response time calculation of a pneumatic brake system. It can be seen from the formula of the derivation of the parameter influence that d and L in the molecule are squared to obtain the unit s, and the influence parameters are integers, which reduces the computational difficulty but limits the computational accuracy of the formula. Thus, the indexes of d, L, C, and p are used as the undetermined coefficient on the basic of the unchanged calculation formula formation as follows:

Here, the undetermined coefficient of the calculation formula is increased and the number of calculation formulas is reduced.

The Levenberg-Marquardt global optimization algorithm is reused to optimize the pressure response time formula coefficients. When the number of iterations is 27, the calculation results are optimal, as shown in Table

Calculation formula.

Application ranges | Formula forms | ||||
---|---|---|---|---|---|

| | ||||

| |||||

Units | Correlation coefficient | Decision coefficient | |||

| |||||

| | | | ||

m | m | m^{3}/(s·Pa) | Pa | 0.99 | 0.99 |

It can be seen from Table

Figure

Analysis of results of improved inflation formula: (a) comparison of experimental and calculation results; (b) calculation errors.

Calculation formula (

Application ranges | Formula forms | ||||
---|---|---|---|---|---|

| | ||||

| |||||

Units | Correlation coefficient | Decision coefficient | |||

| |||||

| | | | ||

m | m | m^{3}/(s·Pa) | Pa | 0.94 | 0.88 |

When

Figure

Calculation error of the formula.

The above formula contains six undetermined coefficients, and the complexity of the formula form is slightly increased, but the probability of improving the accuracy is increased. The Levenberg-Marquardt global optimization algorithm is used to optimize the pressure response time formula coefficients. When the number of iterations is 34, the calculation results are optimal, as shown in Table

Calculation formula (Cg<Cr).

Application ranges | Formula forms | ||||
---|---|---|---|---|---|

| | ||||

| |||||

Units | Correlation coefficient | Decision coefficient | |||

| |||||

| | | | ||

m | m | m^{3}/(s·Pa) | Pa | 0.99 | 0.99 |

The correlation coefficient and the decision coefficient of the formula in Table

Figure

Analysis of results of improved inflation formula: (a) comparison of experimental and calculation results; (b) calculation errors.

The form of the calculation formula is derived based on the dimension analysis method, and the optimization algorithm is used to obtain the undetermined coefficient of the calculation formula; then the calculation formula of the pressure response time is fitted. The calculation formula of the pneumatic line pressure response time is summarized in Table

Calculation formula based on the dimension method.

Application ranges | Formula forms | ||
---|---|---|---|

| | ||

| |||

| | ||

| |||

Units | |||

| |||

| | | |

m | m | m^{3}/(s·Pa) | Pa |

The calculation formula obtained by this method consists of constant term and fractional term. The error of calculation formula is about 8 ms. It can judge the increase and decrease of pressure response time according to the parameter change, and it is easy to determine the parameters of the pipe.

In this paper, under the engineering application of pneumatic brake system, the calculation formula of the pressure response time of pneumatic brake pipe is obtained based on the experiment. The results show that the calculation formula is accurate and has engineering reference value.

Based on the experiment of pneumatic brake pipe, the influence law of each parameter on the pressure response time of pneumatic brake pipe is quantitatively analyzed by fuzzy correlation analysis method. The pipe length is the parameter that has the greatest influence on the pressure response time, and the supply pressure is the parameter that has the least effect on the pressure response time.

The multiparameter variation analysis method is used to analyze the change of pressure response time when the parameters change. The results show that the tube length and supply pressure are positively correlated with the pressure response time of the pipe, and the tube diameter and the inlet sonic conductance are coupled. When the pipe diameter or sonic conductance are in the leading role, the pipe pressure response time is only affected by the dominant parameters, which lays a theoretical foundation for the derivation of the calculation formula.

The calculation formula based on the experimental data directly shows the influence trend of each parameter on the pressure response time, and the formula is simple. By comparing the sonic conductance of the inlet and the pipe, the calculation formula is segmented to ensure the calculation accuracy, which provides a theoretical reference for the design of the pneumatic pipe.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.