A robust optimization design method is proposed to investigate the influence of the hook position on the vertical vibration (bending) of an automobile exhaust system. A block diagram for the robustness analysis of the exhaust system is initially constructed from the major affecting factors. Secondly, the second-order inertia force is set as the vibration excitation source of the exhaust system and the displacement of four hooks of the exhaust system is selected as the variable factor. Then tests are carried out to investigate the resulting vertical bending considering four influencing factors and three levels of analysis. Finally, a variance analysis of the vertical bending is performed. The present study provides a set of guidelines to control the key factors affecting the vibration of vehicle exhaust systems while proposing an effective method to reduce vehicle vibration and improve noise analysis.
The performance of the engine exhaust system not only affects the pumping loss and exhaust emissions but also causes vibration and noise. One end of the exhaust system is connected with the engine, and the other is connected with the vehicle body through a hook. The excitation source transmits the vibration to the exhaust system through the connector, and the vibration is transmitted to the automobile body floor through the hook of the exhaust system. Then, the vibration of the bottom plate is directly transmitted to the passengers through the seat, steering wheel and floor. Meanwhile, the vibration of the vehicle body radiates in the form of noise that causes noise in the car. The automobile exhaust system is an essential environmental protection system that purifies the automobile exhaust, reduces the exhaust vibration and noise, and meets the relevant emission and noise regulations. Moreover, the vibration and noise directly transmitted to the interior of the vehicle through the exhaust system are one of the major factors affecting automobile riding comfort. Therefore, the vibration of the exhaust system is one of the major research topics in automotive NVH (Noise, Vibration, Harshness).
At present, there are few quantitative studies regarding factors affecting automobile vibration. Moreover, the quantitative analysis of the factors affecting the vibration of the exhaust system is even less. Before the vibration control of the exhaust system, it is not enough to understand the contribution degree of each influencing factor to the vibration of the exhaust system only through qualitative research. Because the contribution data of each influencing factor cannot be obtained through research, only judging from the influence degree, there is the possibility of missing the key influencing factors. Consequently, it is necessary to find out the accurate data of each influencing factor’s contribution to the vibration of the exhaust system through quantitative research, in order to avoid missing the key factors of vibration influence and formulate reasonable and feasible measures for the vibration control of the exhaust system. Therefore, in the present study, a robust optimization design method is proposed and a quantitative study of the influence of the hook position movement on the vertical bending vibration of the automobile exhaust system is carried out. In this study, the displacement of the hook position is considered as the main object of the vibration analysis and research of the exhaust system. Moreover, the other secondary factors are ignored, and the second-order inertia force of the engine is used as the excitation source of the exhaust system vibration. The relevant factors affecting the vibration of the exhaust system are investigated to provide a reliable basis for the vibration control of the exhaust system. The proposed method is expected to expand the research of automobile vibration, noise and comfort. Moreover, it is an effective method in improving the accuracy, rationality and effectiveness of automobile vibration and noise control methods.
Many researchers have carried out relevant research. Zhang et al. [
Automobile vibration robustness is defined as vibrations in the process of design, development and operation of automobiles, and the influence of the structure and operating environment on the vibration quality. It is worth noting that the key influencing factors of the vibration can be obtained through the robustness analysis of various influencing factors of the automobile vibration, and the influencing factors can be optimized accordingly. This is called the robustness analysis method.
Theoretically, the reasonable location of the hook can be determined through finite element analysis to achieve the optimal modal frequency of the exhaust system. However, considering the influence of various interference factors in real applications, the optimal position of the hook may change. In this case, it is necessary to study the hook position movement, which has the greatest impact on the modal frequency of the exhaust system. Moreover, the hook position movement should be analyzed to determine the best hook position. Further investigations reveal that many index factors should be considered in the modal analysis of the exhaust system.
Item | Main evaluation indices |
---|---|
1 | Vertical bending mode |
2 | Torsional mode |
3 | Transverse bending mode |
Among the key performance indicators of the exhaust system modal analysis in
The detailed process of establishing the vibration robustness analysis model is as follows:
Define the research object of the model diagram construction. In this study, the research object is the automobile exhaust system.
Analyze the influencing factors of the exhaust system vibration and identify the influencing factors of the exhaust system vibration.
Classify the controllable and uncontrollable influencing factors of the exhaust system vibration. The fluctuation of controllable factors can be controlled by the designer in the vibration target analysis object. In the vibration analysis of the exhaust system, the stiffness of the hook, vibration isolator and flexible connecting pipe are controllable factors. The uncontrollable factors are difficult to determine in the vibration target analysis object and cannot be controlled by the designer. The hook position is an uncontrollable factor in the vibration analysis of the exhaust system.
Identify the vibration excitation source of the exhaust system. In this study, the second-order inertia force is used as the vibration excitation source of the exhaust system, and other excitation sources are ignored.
Define the output variables of the model. In the present study, the output variables are modal frequency and transfer force. Through the analysis of the parameters of these two variables, the quantitative results of the influence on the vibration of the exhaust system are obtained.
Therefore,
Generally, there are four hooks in the exhaust system, and there is a certain deviation between the actual position of the exhaust system hook and the theoretical optimal position. Therefore, quantitative research is carried out to find out the actual optimal position combination of each hook. In order to carry out the quantitative analysis of the exhaust system vibration in a certain range, the hook should move 20 mm before and after the theoretical optimal position. The quantitative influence on the vertical bending modal frequency of the exhaust system is obtained by analyzing the displacement combination of the hook. Therefore, the existing position of each hook has an optimal theoretical position. Moreover, the theoretical optimal position moves backward by 20 mm, and the theoretical optimal position moves forward by 20 mm. Considering the specific situation, only one part of the parameters of the exhaust system is designed. Therefore, the combined data of four hook positions and several different parameters of the exhaust system can be obtained. Then, the data are analyzed by using the robust analysis method. Finally, the quantitative results of the influence of the hook position movement on the vertical bending frequency of the exhaust system are investigated.
The abovementioned analysis shows that the vibration robustness analysis of the exhaust system under the condition of the hook position moving is a four-factor and three-level problem. In the present study, the mutual interference between factors is not considered.
Level number | Factor | |||
---|---|---|---|---|
Hook_{I} | Hook_{J} | Hook_{K} | Hook_{L} | |
1 | Move forward 20 cm | Move forward 20 cm | Move forward 20 cm | Move forward 20 cm |
2 | Best position | Best position | Best position | Best position |
3 | Move back 20 cm | Move back 20 cm | Move back 20 cm | Move back 20 cm |
The robust method is an optimal design method of using an orthogonal array to arrange and analyze a variety of factors. A representative horizontal combination from all of the combinatorial tests is carried out. Through the experimental results, the overall test situation and the optimal design method of the level combination are obtained. It should be indicated that this method has strong orthogonality, representativeness and comprehensiveness.
Since mutual influence between the hook position movements is not considered, the orthogonal table
experimental serial number | Serial number | experimental design | ||||||
---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | |||||
1 | 1 | 1 | 1 | 1 | I_{1} | J_{1} | K_{1} | L _{1} |
2 | 1 | 2 | 2 | 2 | I_{1} | J_{2} | K _{2} | L _{2} |
3 | 1 | 3 | 3 | 3 | I_{1} | J_{3} | K _{3} | L _{3} |
4 | 2 | 1 | 2 | 3 | I_{2} | J _{1} | K _{2} | L _{3} |
5 | 2 | 2 | 3 | 1 | I_{2} | J _{2} | K _{3} | L _{1} |
6 | 2 | 3 | 1 | 2 | I_{2} | J _{3} | K _{1} | L _{2} |
7 | 3 | 1 | 3 | 2 | I_{3} | J _{1} | K _{3} | L _{2} |
8 | 3 | 2 | 1 | 3 | I_{3} | J _{2} | K _{1} | L _{3} |
9 | 3 | 3 | 2 | 1 | I_{3} | J _{3} | K _{2} | L_{1} |
Four major factors that affect the vibration of the exhaust system are as the following: 1) The engine connected with the exhaust system, and the mechanical vibration generated by the engine is directly transmitted to the exhaust system. 2) The vibration of the tire and vehicle body caused by road excitation results in the vibration of the exhaust system. 3) The vibration caused by the rapid airflow generated by engine combustion in the exhaust system pipe. 4) The hook is connected with the car body, which causes the vibration of the exhaust system when the automobile body vibrates. Considering the primary and secondary correlation of excitation sources and model building factors, only the vibration caused by the engine is analyzed. According to the cause and dynamic analysis of the engine excitation force, the second-order inertia excitation of the engine is mainly considered. The excitation model can be mathematically expressed as follows:
where P and λ denote the second reciprocating inertia force and the connecting rod ratio, respectively. Moreover, R and
Since the vibration energy of the vehicle is mainly concentrated in the frequency band of 20Hz–125Hz, the frequency spectrum of the second-order reciprocating inertia harmonic excitation with the engine speed between 800 r/min and 4000 R/min is analyzed by the Fourier transform. Then,
In order to investigate the influence of the hook position movement on the vertical bending vibration, the average response value of the level number of each hook position movement should be initially calculated and the corresponding response diagram should be drawn according to the average response value. According to the correlation diagram of the results between the hook position movement and the experimental design, the influence trend on the vertical bending vibration of the exhaust system can be obtained.
Serial number | Factor | Vertical bending frequency results | |||
---|---|---|---|---|---|
Hook_{I} | Hook_{J} | Hook_{K} | Hook_{L} | ||
1 | 1 | 1 | 1 | 1 | 28.1 Hz |
2 | 1 | 2 | 2 | 2 | 29.4 Hz |
3 | 1 | 3 | 3 | 3 | 30.3 Hz |
4 | 2 | 1 | 2 | 3 | 31.6 Hz |
5 | 2 | 2 | 3 | 1 | 32.7 Hz |
6 | 2 | 3 | 1 | 2 | 30.2 Hz |
7 | 3 | 1 | 3 | 2 | 32.1 Hz |
8 | 3 | 2 | 1 | 3 | 29.5 Hz |
9 | 3 | 3 | 2 | 1 | 31.4 Hz |
The calculation formula of the level sum of hook_{I} is expressed as follows:
The level sum of hook_{I} is as follows:
The level sum of other hooks is as follows:
The equation of the level mean value of hook_{I} is as follows:
The level mean value of hook_{I} is:
The level mean value of other hooks is as follows:
Considering the calculation of the level value and level sum mean value, the influence trend chart is drawn to show the concentration trend, fluctuation degree and distribution shape of the test data in the scheme. Therefore, the results of the influence of various factors on the vibration of the exhaust system are obtained.
According to the experimental results of each hook, the influence trend diagram of each factor can be drawn. x-coordinate and y-coordinate present the level value and the level sum mean value, respectively. Then, the coordinate points formed by each level value and level mean values are connected with approximate straight lines.
It is observed that different levels and level mean values of the same parameter reflect the trend of the influence on the vertical bending vibration of the exhaust system. Moreover, it is found that the movement of hook_{I} and hook_{L} has an obvious effect on the vertical bending vibration of the exhaust system, while hook_{J} and hook_{K} do not have an obvious effect on the vertical bending vibration of the exhaust system.
The equation for calculating the range of affecting factors is as follows:
The range of hook_{I} is calculated as follows:
The range of other hooks is calculated as follows:
According to the abovementioned equations, the ranges of the hooks in descending order are as follows:
It is worth noting that the smaller the hook range, the smaller the influence of the hook movement on the vertical bending modal of the exhaust system. Moreover, the greater the hook range, the greater the influence of the hook movement on the vertical bending mode of the exhaust system. Therefore, it is necessary to be quite careful when adjusting the hook position with a large range value; otherwise, the vibration will be strengthened. Therefore, the movement of hook_{K} has the biggest impact on the vertical bending mode of the exhaust system, followed by hook_{I}. On the other hand, the movement of hook_{J} has the least impact on the vertical bending vibration of the exhaust system, which means the best effect on the vertical bending vibration robustness of the exhaust system, followed by hook_{L}. Therefore, the vibration robustness analysis under the condition of four hooks as uncontrollable factors shows that the position movement of hook_{K} and hook_{I} should be reduced as far as possible to improve the robustness of the vertical bending vibration of the exhaust system. However, the influence of hook_{J} and hook_{L} position movement on the vertical bending vibration of the exhaust system can be ignored.
Considering the variance analysis of orthogonal experimental data of the hook position movement, the contribution rate and the primary and secondary correlation of the impact of hook position movement on the vertical bending vibration of the exhaust system can be obtained quantitatively and accurately. Therefore, in order to obtain an economical and practical design scheme, the key factors that affect the vibration should be adjusted and controlled appropriately, while the secondary factors should be ignored.
Total square sum can be calculated through the following expression:
In the abovementioned equation
The calculation equation of the total degree of freedom is as follows:
The total degree of freedom is calculated as follows:
The equation of single factor sum of square is as follows:
The single factor sum of square of hook_{I} is calculated as follows:
The single factor sum of square of other hooks is calculated as follows:
Single-factor degrees f freedom are as follows:
where n refers to the number of levels with different values of influencing factors of the exhaust system vibration. In this study, the value of the level number is 3, so n = 3
The single factor degrees of freedom are calculated as follows:
Since there are no empty column items in the orthogonal experiment scheme, the sum of S_{J} of hook_{J} and S_{L} of hook_{L}, which has a smaller sum of square fluctuations of influencing factors in the orthogonal test, can be taken as Se. Moreover, the sum of f_{J} of hook_{J} and f_{L} of hook_{L} can be considered as error fluctuation square sum degree of freedom
Source of deviation | Square sum S | Freedom f | Mean square V | Statistic F | Contribution ratio% ρ |
---|---|---|---|---|---|
I | 8.42 | 2 | 4.21 | 129.54 | 46.0 |
J | 0.02 | 2 | 0.01 | 0.33 | |
K | 8.08 | 2 | 4.04 | 134.67 | 47.9 |
L | 0.11 | 2 | 0.06 | 2 | |
e | 0.13 | 4 | 0.03 | 6.1 | |
Total square sum | 17.45 | 8 | 100 |
The Intuitive analysis gives the qualitative analysis of the influence of the hook position movement on the vertical bending mode of the exhaust system. On the other hand, the variance analysis gives the quantitative conclusion of the influence of the hook position movement on the vertical bending mode of the exhaust system, which provides an important judgment basis for the control of the noise factors under the hook position movement.
In the present study, influencing factors of the exhaust system are investigated by the robust optimization design method. Based on the robust analysis of influencing factors of the exhaust system, this study investigates the vertical bending modal of the exhaust system with the hook position moving from qualitative and quantitative aspects. A schematic diagram for the robustness analysis of the vertical bending vibration of the exhaust system with four moving hook positions is constructed. The second-order inertia force of the engine is the excitation source of the exhaust system, and the four hook positions of the exhaust system are taken as variable factors. Moreover, using the
The robust optimization design is used to study the vertical bending vibration of the automobile exhaust system. Therefore, not only the qualitative results of the influence of the hook position movement on the vertical bending vibration modal of the exhaust system can be obtained, but also the quantitative data of the influence of the hook position movement on the vertical bending vibration of the exhaust system can be obtained. The proposed method is beneficial for the control of key factors of the vehicle exhaust system vibration. Moreover, it has a certain reference for the development of vehicle vibration and noise analysis methods. It is worth noting that when using the proposed method to study the related issues, attention must be paid to the analysis of the key influencing factors. Otherwise, omission problems occur in the process of the system robustness optimization, which will lead to a large error of the method analysis results and affect the accuracy of the analysis method conclusions. The level number of influencing factors should be determined according to the actual situation. Too many levels lead to an increase in the number of experiments, which will lead to an increase in the calculation amount of the method. On the other hand, too few levels lead to the unreliability of the experimental data. In the follow-up study of the method, the complexity, time-consuming level and related software system development of the method should be considered to improve the applicability of the method and improve the efficiency and convenience of the algorithm.
The authors would like to thank the anonymous reviewers for the helpful suggestion.