In this study, a finite-element simulation model was established for a two-dimensional hydraulic cylinder seal structure with highly nonlinear materials and contacts for achieving the optimal sealing effect with the structure. The effects of gaps as well as single and double O-rings on the sealing effect were examined. On the basis of this examination, a parametric model was developed for double-O-rings without gaps, which are suitable for hydraulic cylinder sealing. Suitable design variables, objective functions, and constraint conditions were determined for the parametric model. Then, a surrogate model was fitted and optimised through a constrained Latin hypercube method, an interpolating recmultiquadric radial basis function method, and the genetic algorithm. The results indicate that the seal and seal groove structures obtained through optimisation with the surrogate model provide a superior sealing effect to unoptimised structures. Therefore, the combination of the developed surrogate model and finite-element method can provide a theoretical reference for the design of the sealing structure of hydraulic cylinders.
Hydraulic transmission has attracted considerable attention for transportation vehicles, such as automobiles, ships, and aircraft, due to the development of manufacturing with the advancement of technology. In a hydraulic system, structures such as valves, pumps, and cylinders are connected to hydraulic pipe joints through a piping system. Hydraulic pipe joints, pumps, valves, and cylinders are the most basic hydraulic system components that play a vital role in the circulation of gas, water, and oil in the system [
Currently, the research on and production capacity of seals in China is relatively low, and manufacturers blindly increase the compression amount of seals in most cases to solve the problem of hydraulic seal failure [
In general, the functional components of a sealing device are the upper and lower flanges and the middle sealing rubber. Therefore, a study of the sealing performance should begin from the analysis of these three components. The amount of compression of the sealing rubber directly affects the sealing performance of the hydraulic cylinder sealing structure. Moreover, large compression reduces the service life of the sealing rubber. The cross-sectional shape of the sealing rubber directly affects the contact area between the sealing material and the upper and lower flanges on the application of the pretightening force and thus affects the sealing efficiency. The structures of the upper and lower flanges also affect the tightness of the hydraulic cylinder sealing structure. The size of the gap in the groove of the seal ring affects the amount of deformation of the seal rubber, which affects the sealing ability of the device. Chen et al. obtained an O-type seal with a higher reliability than the contact pressure distribution between the rectangular seal and the O-ring [
This study mainly used Ansys Workbench to perform finite-element analysis on the seals of hydraulic cylinders, compare the seal ring section shape, flange groove width, and groove depth, and optimise the groove structure through a surrogate model in MATLAB for obtaining a set of optimal structural parameters for the sealing rubber and flange and for achieving the best sealing effect for the hydraulic cylinder sealing structure.
The seal of a hydraulic cylinder has a typical symmetric structure. Considering the symmetry of the structures of the seal ring and hydraulic cylinder, this study simplified the three-dimensional plane strain problem for a hydraulic cylinder into a two-dimensional problem for simulation calculation. The two-dimensional axial seal models of the four seals considered in this study, which are based on the Chinese standard GB/T-3452.3-2005 [
Two-dimensional models of the four seals considered in this study.
Figures The rubber material is an isotropic and uniform material. The creep properties of rubber materials are uniform during elongation and compression, and the volume change is negligible when creep occurs. Compared with hydraulic cylinders, the quality of O-type rubber seals is negligible. The overall model of the hydraulic cylinder can be simplified into two dimensions.
Because the sealing ring is usually made of rubber material and has a high degree of nonlinearity in the finite-element simulation, this study adopted the Mooney–Rivlin model with two parameters to represent the constitutive relationship of the material with large deformation [
The aforementioned formula indicates that the mechanical property constant of a material can be obtained from the Hs value and
Considering the oil and water resistance, NBR is selected for analysing many rubber materials. The hardness of the seal under high pressure in this study was 85. According to the aforementioned formula,
Network segmentation is a critical step when a model is used for numerical simulation analysis. They must be as regular as possible in addition to the number of units that need to be controlled. Therefore, element independence analysis is usually required before determining the element size. As displayed in Figure
Element independence verification.
Due to the large friction forces between the sealing rubber and the upper and lower flanges, the contact method was set to frictional contact, and the friction coefficient was 2 × 10−2. The contact algorithm uses the augmented Lagrange multiplier method to adapt to large-deformation contact problems. Moreover, the finite-element control method uses the integration point detection method (Gauss Point) to analyse the friction contact problem [
In the finite-element analysis of the seal in this study, a 5.3 mm-diameter nitrile rubber was used as the seal ring. The width B of the seal groove with a gap was 6.3 mm, and the width B of the seal groove without a gap was 5.3 mm. The seal groove depth of the single groove was 4.13 mm. The first force causes large deformation, and second force constrains all the displacements of the lower flange. Three load steps were set. The first load step was the initial situation without any force; the second load step involved the application of a pretightening load, and the compression was 20% of the seal ring (i.e., 1.06 mm); and the third load step involved setting the internal pressure of the hydraulic cylinder to 2 MPa. In the simulation, the left side was assumed as the inside of the hydraulic cylinder and the right side was assumed as the external side. Therefore, a pressure load of 2 MPa was applied to the left side of the seal.
A comparison of the results obtained for a single-groove seal with and without gaps is displayed in Figure
Equivalent stress and contact pressure cloud diagrams of the single-groove seal.
Figures
Mean average pressure and maximum contact pressure for different sealing methods.
Single seal with gap (MPa) | Single seal with gapless (MPa) | Double seal with gap (MPa) | Double seal with gapless (MPa) | |
Mean equivalent stress | 2.66 | 2.66 | 2.65 | 2.65 |
Maximum contact pressure | 4.3808 | 4.3818 | 4.3876 | 4.3888 |
The contact pressure is the major factor affecting the sealing effect. When the contact pressures are greater than the pressures in the hydraulic cylinder, the sealing is successful and the sealing effect is proportional to the contact pressures. A comparison of Figures
Structural optimisation design involves determining a set of optimal solutions from design variables under the proposed constraints. The general optimisation solution process involves mathematically modelling the optimisation problem, proposing constraints according to the mathematical model, and selecting an appropriate method to optimise the established model for a certain goal. In general, the following parameters are considered in the process of optimal design [ Design variables: design variables are parameters that have a major influence on the model (design variables are generally denoted by Objective function: the objective function is the optimisation objective (optimisation object), which is generally denoted as Constraints: constraints refer to the ranges of design variables. The types of constraints in the optimisation process include stress and volume constraints.
According to the simulation results presented in Section
In this study, the maximum value of the contact pressure between the seal ring and the upper flange was used to evaluate the sealing effect [
The current diameter of the seal ring was 5.3 mm, and the diameter of the seal ring (
The Latin hypercube method [
The number of test points was selected according to the following principle: the number of test points must be 10 times higher than the number of variables. Two design variables were used in this study; therefore, 20 points were selected to build the surrogate model. The plane distribution of the design variables is displayed in Figure
Plane distribution of design variables.
After the selection of the test points, the maximum contact pressure between the sealing ring and the upper flange was calculated for all the training and test points through finite-element simulation. Ansys Workbench was directly used for parametric modelling and finite-element calculation. The corresponding results are presented in Table
Data points and response values.
5.14 | 3.39 | 6.528 |
6.23 | 3.98 | 4.1881 |
4.54 | 3.26 | 4.8721 |
5.28 | 3.45 | 6.4509 |
6.31 | 3.88 | 4.1706 |
6.85 | 4.49 | 4.027 |
5.77 | 3.34 | 4.3518 |
5.85 | 4.57 | 4.302 |
5.63 | 4.18 | 4.3994 |
6.92 | 4.38 | 4.015 |
5.00 | 3.18 | 6.3237 |
6.62 | 4.13 | 4.0882 |
6.15 | 3.55 | 4.2387 |
6.69 | 4.34 | 4.0658 |
6.08 | 3.95 | 4.234 |
7.00 | 5.79 | 3.992574 |
4.30 | 2.95 | 5.0262 |
6.54 | 5.03 | 4.1055 |
6.38 | 4.95 | 4.1484 |
4.79 | 3.49 | 6.5858 |
The interpolating recmultiquadric radial basis function method [
Response surface and residuals of the training points.
The residual error of the training points was 10−7 (Figure
The genetic algorithm [
Optimal design variables and optimal solutions.
4.23 | 2.8 | 7.2778 |
Let
However, the structure function should be standardized in the actual calculations:
The probability of failure after standardization can be expressed as
The reliability of the structure can be expressed as
In this paper, the Gram–Charlier series was used to decompose and fit the distribution of the function, which could be expressed as
According to the finite-element analysis results above, the mean values and the mean square error in case of the maximum contact pressure after storage and aging of the sealed structure can be calculated. The results are shown in Table
The mean values and the mean square error in case of the maximum contact pressure after storage and aging of the sealed structure.
Time of usage | Shortly after sealing | 10 years | 20 years | 25 years | 30 years |
Mean of response (MPa) | 4.59 | 4.57 | 4.38 | 3.72 | 3.64 |
Mean square error of response (MPa) | 0.94 | 0.88 | 0.84 | 0.72 | 0.72 |
According to the finite-element simulation and the surrogate model, the limit state equation of the sealing structure was established, and the corresponding moments of the structure under the maximum contact pressure in each aging period can be obtained, and the reliability of the sealing structure can be calculated according to the above equations. In this paper, five nodes in the middle of the O-ring were selected for calculation. The results of reliability for the sealing structure joints are shown in Table
Reliability of 5 sealed structure joints.
Serial number | Shortly after sealing | 10 years | 20 years | 25 years | 30 years |
1 | 0.976 | 0.921 | 0.851 | 0.827 | 0.796 |
2 | 0.983 | 0.926 | 0.846 | 0.821 | 0.793 |
3 | 0.984 | 0.927 | 0.859 | 0.835 | 0.794 |
4 | 0.987 | 0.931 | 0.861 | 0.845 | 0.807 |
5 | 0.994 | 0.938 | 0.881 | 0.848 | 0.813 |
The average value of the five nodes was used to represent the reliability of the sealing structure; it can be seen that the reliability of the sealing structure after 10 years, 20 years, 25 years, and 30 years was 0.985, 0.929, 0.859, and 0.801, respectively. A failure rate less than 10% is acceptable for ordinary hydraulic cylinder sealing structures. Therefore, the sealing structure could still be effective after 10 years.
The parameters obtained after design variable optimisation were substituted into the model, and simulation was performed using Ansys Workbench. The maximum contact pressure was obtained as 7.2794 MPa. The error between the aforementioned value and the corresponding result obtained using the surrogate model was 0.027%. The model and simulation results are presented in Figure
Model and simulation results.
Figure
Comparison of the optimised and unoptimised results.
Unoptimized seal structure (MPa) | Optimized sealing structure (MPa) | Lift percentage (%) | |
Mean equivalent stress | 2.65 | 5.01 | 89.06 |
Maximum contact pressure | 4.39 | 7.28 | 65.83 |
Maximum shear stress | 2.67 | 2.65 | −0.75 |
According to the aforementioned simulation results, all the structures, the O-ring seal did not contact the chamfers of the seal grooves under a preload of 1.06 mm and working pressure of 2 MPa. The equivalent stress had a spindle-shaped distribution and was larger at the contact area than at other areas. The average equivalent stress of the optimised structure was 89.06% higher than that of the unoptimised structure. Moreover, the maximum contact pressure for the optimised structure was 65.83% higher than that for the unoptimised structure. The maximum shear stress indicates the service life of the seal ring. The maximum shear stress of the optimised seal structure was 0.75% lower than that of the unoptimised structure; thus, the shear stress was only marginally reduced after optimisation. The aforementioned results indicate that the optimised structure has a considerably stronger sealing effect and only marginally smaller service life than the original structure.
The obtained qualitative finite-element simulation results indicate that, during the operation of hydraulic cylinders, the peak values of the equivalent stress and contact pressure are always located at the contact between the seal ring and the upper flange in the prestressed state. Thus, the sealing effect of a hydraulic cylinder seal can be assessed according to the maximum contact pressure at the contact between the seal and the upper flange. The finite-element analysis results obtained for the four sealing structures considered in this study indicated that the sealing effect of the gapless double O-ring was stronger than those of the double O-ring with a gap, the gapless single O-ring, and the single O-ring with a gap. According to the surrogate model and finite-element simulation calculations, the contact pressure of the optimised seal structure is 65.83% higher than that of the original gapless double O-ring. The combination of the developed surrogate model and finite-element method can provide ideas for the optimal design of seal structures without sacrificing the service life.
The datasets supporting the conclusions of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This project was supported by National Natural Science Foundation of China (Grant no. 51975092)