Research of the Bending characteristics of Aero Ball Joints at High Temperature and Pressure

As a piping element with excellent stress compensation performance, spherical joints are widely used in aerospace piping systems. However, in piping stress simulation, the presence of nonlinear features such as friction vice of the solid model of the spherical joints will lead to the difficulty of convergence of the calculations or the deviation of the results is too large, so in order to improve the convergence of the iterative process, “joint” joints are used to replace and simplify the original spherical joints, but this simplified calculation needs to accurately capture the characteristic parameters (such as bending stiffness and bending friction moment, etc). In order to improve the convergence of the iterative process, “joint” is usually used to replace and simplify the original spherical joint, but this simplified calculation needs to accurately capture the characteristic parameters of the spherical joint (e.g. bending stiffness and bending friction moment, etc.). This paper focuses on the bending characteristics of spherical joints by establishing a high-temperature and high-pressure test platform, and takes into account the difference between the test and the actual use of spherical joints, and further combines numerical simulation and BP neural network methods to predict the characteristics of spherical joints to meet the requirements of the actual use of spherical joints.


Introduction
Spherical compensators have been widely used in piping systems due to their ease of installation, low thrust at the fixing point and ability to provide large compensation in the piping system, as well as their superiority in terms of economy, safety and durability.
Spherical compensator can provide compensation for the pipe system within a certain angle of the angle, so it is widely used in industry.Mamoru Kitahara [1] proposed a calculation method for the use of spherical compensator and pointed out that the amount of expansion of the pipe system as well as the load of the fixed bracket and guiding bracket are the main reference factors for the selection of the installation position of the spherical compensator.Jin Kunyu [2] proposed an equivalent alternative method for the simulation calculation of spherical compensator pipeline, firstly, zero-length expansion joints can be used for simulation, and secondly, constraints and connection points can also be used for simulation.Li Guanghui [3] summarized the two most common forms of spherical joint arrangement in industrial applications: the first is the arrangement of three spherical compensators on the L-shaped pipeline, installed in the length of the short and long sections of the pipe section to absorb the displacement, and the second is the arrangement of two spherical compensators on the vertical or horizontal Z-shaped pipeline, the mechanism is similar to that of the square compensator but it can be used for maintenance without stopping the production of pipelines under pressure.Cai Xin [4] on the use of pipeline thermal compensation principles and installation precautions, proposed cold tight, selfcooled tight and cold tight ratio selection is an important criterion for the selection of spherical compensator, in the allowed circumstances should be chosen to increase the actual amount of In modern industrial applications, scholars have made many improvements to this structure.Huang Zhongtai [7] summarized and compared the cold-pressure type corrugated compensator and coldpressure type outer covering ring corrugated compensator, the former is suitable for normal temperature and high pressure pipeline in practical engineering applications, while the latter is suitable for high temperature and high pressure pipeline engineering projects.Tang Yujian et al [8] proposed a method of calculating the static pressure thrust borne by the corrugated compensator in the pipeline design, and at the same time, a new type of corrugated compensator can be used in the large diameter and high pressure pipeline, which can be used for balancing the static pressure thrust within a certain value of the pipeline in which the compensator is located.In response to the excessive hydrostatic thrust, Ni Hongqi et al [9] developed a fault diagnosis system of corrugated compensator for accident early warning.On this basis, Su Jie [10] proposed the influence of blind force on the selection of corrugated compensators, and this influence factor is not generally considered in the natural compensation of the pipe system.Li Qiu et al [11] carried out the design of balanced corrugated compensator, and based on the engineering application of experimental verification, put forward the engineering application of matters needing attention.In engineering in recent years for the corrugated compensator displacement monitoring has more research and attention, Wang Ziyang et al [12] designed a use of linear displacement sensors to measure the displacement generated during the work of the compensator, and through the physical test to verify the accuracy of the design as well as realtime.Based on the shortcomings of the terminal monitoring system, Ni Hongqi et al [13] designed a compensator monitoring system that can be applied to mobile terminals, and sent the data to the server stably by piggybacking on wireless communication technology.After that, Dong Xiaofang [14] further improved the wireless monitoring system in terms of development, increased the data processing function at the acquisition end, and optimized the compensator displacement monitoring system.

Structural Composition and the Compensation Principle of Ball Joints
Both simulation calculations and theoretical studies generally require a certain time period, and cannot provide stress state monitoring and assembly solutions in the pipe installation and use stages, so it is necessary to conduct stress tests about pipelines, which is of great significance for the progress of actual pipeline analysis technology in the industry.How to improve the accuracy of stress tests has become one of the popular research directions for scholars and engineers in recent years.
In 2019, Feng Xin et al [15] proposed a high-temperature pipeline stress monitoring method and conducted a prototype test, where the stress sensing element is a fiber optic sensor distributed based on the shape of the pipeline, which is mainly based on the interrelationships between axial and bending stresses, as well as pipeline thermal deformation, bending deformation, and other mechanical mechanisms, and then the strain monitoring data are used to calculate the axial as well as circumferential strain data, and then the real-time The structural safety performance of hightemperature pipeline stress is evaluated.
In the same year, Yu Weigang et al [16] proposed a test method for the numerical calculation of pipeline installation stresses, and after the pipeline installation stress test bench had been built, the stress-related test was carried out on the pipeline by the hammering method [17], which investigated the relationship between the stresses, the modal parameters and the pipeline characteristics, and verified the method by the test data.Liao [18] for nuclear piping system stress analysis using shell and tube unit for modeling, for the stress concentration of the location of the local refinement to obtain more accurate data for the global mechanical model of the piping system to carry out research.
In 2020, Lv Jing et al [19] analyzed the weak magnetic effect of materials under stress using the energy balance theory of ferromagnetic materials.The main advantage of this method is that there is no need to magnetize the pipeline under test, and it has the advantages of simple equipment operation, large monitoring coverage, and short detection period, etc.The existing engineering experience was borrowed for the detection of the common typical stress concentration pipeline segments in the pipeline system, and then compared and analyzed the results with the rest of the detection techniques (ultrasonic, electrical resistance).In 2021, Xiang Chaoyuan et al [20] improved the above method, hardware with high-precision Hall sensors for data acquisition, can detect the remanent magnetization signal on the surface of the magnetized plate with curvature, and at the same time for the stress on the metal pipeline caused by the signal changes caused by the change in the data changes are analyzed and considered for the development and application of non-destructive testing technology for the pipeline to provide a reference.In this paper, the test method is to synthesize the above research status for the test design, data acquisition, and combined with the characteristics of the test instrument, simulation data results for the collected stress values for error analysis and simulation results of the calculation of the control, to provide practical theoretical references for the analysis of pipeline stress compensation.Spherical joints, i.e. spherical bellows compensators, mainly rely on the expansion and contraction of the bellows to release the stress, used for thermal stress compensation and to increase the flexibility of the pipeline, belong to the non-detachable pipeline components, the use and installation of the spherical joints should be considered when the following factors: The angle of bending has a linear relationship with the bending moment as shown below.

Test platform
The size of the spherical joint studied in this paper's test was 2 inches and the material was GH625 with a density (8442.37 kg/m3), tensile strength of 820.5 MPa, and yield strength of 386.1 MPa.The test required the establishment of a high-temperature and high-pressure test rig to obtain the bending characteristics of the spherical joint.The test was accomplished on an electro-hydraulic servo testing machine, as shown in the figure below, where straight tubes were welded at both ends of the spherical joint to facilitate the installation of the test article.The left-hand side of the tube was mounted on a support, while the right-hand side of the port was mounted in a linear bearing that moved vertically and allowed the tube to slide inside.The displacement load of the linear bearing is controlled by the testing machine.The test piece is heated in a furnace and sealed at both ends to allow pressurization by means of a high-pressure gas cylinder.The material of the specimen was GH4169.In addition, different pressure conditions also change the rigidity, so capturing the motion and compensation characteristics of a spherical joint under different airflow parameters is of great research as well as practical significance for the thermal compensation design of spherical joints.This paper focuses on the compensation characteristics of a ball joint under three different temperatures and pressures.

Experimental Results
The deflection angle-moment curves are shown in Figure 5 below.The characteristic parameters are listed in Table 3.As shown in the table above, the bending friction moment increases significantly with increasing pressure.The moments and stiffness decrease with increasing temperature, this is because the change in temperature leads to a decrease in the G modulus of elasticity, which makes the specimen more susceptible to deformation.
From the above study, although the temperature has a certain effect on the bending friction moment and bending stiffness of the spherical joints, when the temperature is increased from 220 ℃, the change of bending friction moment under different pressures is 5.46 %, 5.68 % and 5.89 %, respectively, and the change of bending stiffness is 16.28 %, 18.59 % and 23.91 %.When the pressure is increased from 1 MPa, the changes of bending friction moment at different temperatures are 83.56%,82.13% and 82.79%, and the changes of bending stiffness are 61.68%,53.11% and 46.29%, respectively.Therefore, the compensation characteristics of spherical joints are more significantly affected by pressure.
The following figure gives a cloud diagram of the stress distribution of the inner ball strand and bellows under one working condition, at this time, the inner and outer ball strand fit more "tightly", the structural rigidity increases, which increases the bending friction moment and bending stiffness.

Correction Method Introduction
In order to obtain the characteristic parameters of the ball joint under gas flow in the tube, firstly, it is necessary to carry out tests and finite element analysis.Secondly, the data obtained from test and finite element analysis are compared and analyzed with error correction.A BP artificial neural network model is established to analyze the finite element results of the spherical joint end without pressure, and the characteristic parameters of the spherical joint are given.

FE Analysis of a Ball Joint
The FE meshed model is shown in Figure 7.According to the actual test conditions, under the full consideration of the influence of the main parameters, one end of the test piece model is fixedly restrained, the other end is subjected to a tensile force along the opposite direction of gravity, while the pressure load is applied to the pipeline and the inner wall of the spherical joints, and the overall model is warmed up in order to apply temperature loads.
The deflection characteristics of the spherical joint were analyzed using the finite element method, as shown in Fig.The finite element analysis results are in obvious trend consistency with the test results, but unlike the test results of the spherical joint characteristics, the curve of the finite element calculation results of the spherical joint is smoother, with no obvious bending friction torque points, and the reason for this phenomenon is that the simulation and analysis process did not consider the GH625 friction vice under different working conditions.The reason for this phenomenon is that the simulation analysis process did not consider the change of friction coefficient of GH625 friction pair under different working conditions; The deflection characteristic curve of the spherical joint without pressure at the end was also analyzed using the finite element method, as shown in Figure 9 below.

Establishment of the BP ANN Model
In this paper, an artificial neural network model was developed using Matlab software to investigate the relative error under different operating conditions.The input layer of the network model has 3 nodes, which are temperature, pressure and deflection angle.After calculation and analysis, the number of nodes in the implicit layer is selected as 12.The output layer has 1 node, and the final network structure is 3×12×1.
In each group of finite element analysis and test of the relative error and deflection angle of the curve, select the data points to establish the network, the data points used for network training from the above overall data randomly selected 80 %, as follows: Neural network numerical comparisons are shown in the figure and can be used for data prediction of research results.

Results and Analysis
According to the results of finite element analysis, it is found that the deflection angle corresponding to the bending friction torque is reduced by 0.4~0.8° in the case of gas flow compared with that in the case of no gas flow, and the deflection characteristic curves are corrected for 0~5° as shown in Fig.
The corrected results for 0~3° are distorted, while the results for 3~5° are closer to the linearity.The final correction curve is the black curve in Figure 11.The bending properties under different loads were corrected, as shown in Figure 12.From the data in the table, it can be seen that the rigidity of the spherical joint in the pipe system is much smaller than the result in the monolithic test (gas does not flow), and the bending friction torque is reduced by 77.91% under the pressure of 2 MPa and the temperature of 220 ℃.The data show that the method proposed in this paper can effectively correct the characteristic parameters of the spherical joint in the actual working condition, which provides a reference and support for the design of hightemperature pressure pipeline and the compensation of stresses.The data show that the method proposed in this paper can effectively modify the characteristic parameters of spherical joints in practical working conditions, which provides reference and support for the design of high temperature pressure pipelines and stress compensation.

Conclusions
In this paper, the characteristics of spherical joints under different working parameters are studied by the designed test bench for testing the deflection characteristics of high temperature spherical compensators.On this basis, finite element analysis of the deflection characteristics of the spherical joints under the same working conditions of the test was carried out, and the measurement error of the spherical joint test was analyzed in comparison with the test results.Thirdly, a neural network model was constructed, and a correction method for the deflection characteristic curve of the spherical joint was proposed, and the characteristic curve and parameters under the internal gas flow condition were obtained.The specific research conclusions are as follows: (1) In the deflection process, the spherical joint has nonlinear characteristics, with "high stiffness" before the moment reaches the bending friction moment, and "low stiffness" after the moment reaches the bending friction moment; (2) The greater the pressure, the greater the rigidity of the same text ball joint, bending friction moment and bending stiffness with the increase in pressure, when the pressure increases from 1 MPa to 2 MPa, the bending friction moment increment reaches 83.56%, bending stiffness increment reaches 61.68%.When the pressure is constant, the bending friction moment and bending stiffness decrease with the increase of temperature due to the change of physical parameters of the material at high temperature.The key parameters of the compensation characteristics of the spherical joint are more obviously affected by the pressure.
(3) The results predicted by the neural network model with temperature, pressure, and deflection angle as inputs are in high agreement, and the correction method of the deflection curve of the spherical joints based on the research content provides a reference and support for the design of hightemperature pressure pipelines and stress compensation.

1 .Figure 1 .
Figure 1.Schematic illustration of a ball joint: (a) sectional view; (b) structure of the ball joint.

Figure 2 .
Figure 2. Ideal bending characteristics of a ball joint.

Figure 4 .
Figure 4. Schematic diagram of the experiment.

Figure 5 .
Figure 5. Bending characteristic curves of ball joints under different loads.

Figure 8 .
Figure 8.The relative error of experiment results and FE results

Figure 9 .
Figure 9. Bending characteristics of the ball joint without the ends blocked.

Figure 10 .
Figure 10.Comparison of predicted values and expected values of the BP neural network test set.

Table 1 .
Degrees of freedom of a ball joint in different directions.

Table 2 .
Material properties of GH4169 at different temperatures.
3.2.Experimental ContentMaterial properties of spherical joints such as coefficient of linear expansion change with temperature.

Table 3 .
Characteristic parameters of ball joint under different loads.

Table 4 .
Correction value of bending characteristics.