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In recent years, hydropneumatic suspension (HPS) has come into widespread use for improving the ride comfort and handling of mining dump trucks and off-road vehicles. Therefore, it is critical to improve the mathematical modeling accuracy to enhance the design and control efficiency and accuracy of HPS. This paper aims to propose a model for improving the modeling precision by considering the effect of different factors on HPS characteristics. A computational fluid dynamic (CFD) model of a HPS was developed, and the volume of fluid (VOF) method was used for the transient calculations in order to simulate the fluid dynamic characteristics and determine the damping and stiffness forces of HPS. The effect of temperature, oil viscosity, nitrogen dissolution rate, and suspension vibration speed on the nonlinear characteristics of HPS was investigated. A limited number of simulation sample points were designed based on the variation ranges of the above factors using the design of experiment (DOE) method. The corresponding damping and stiffness force of each sample point were calculated by CFD simulation. The obtained simulation data were utilized for the fitting of a Kriging model. The results demonstrated that the Kriging model can provide high accuracy, with a prediction error lower than 5%. The proposed modeling method of the HPS nonlinear characteristics is highly efficient, accurate, and faster than traditional methods.

Due to their nonlinear stiffness and damping characteristics, hydropneumatic suspensions (HPSs) are able to attenuate the body vibration and impact caused by road unevenness in a transitory period, improving handling stability and ride comfort [

Most of the vibration energy absorbed by the suspension system is dissipated in the form of thermal energy, which leads to the increase of temperature in the HPS. Thermodynamic models have always been used to evaluate the temperature increment [

At present, the ideal and real gas law approaches have been employed to calculate the effect of HPS gas state change [

Furthermore, hydraulic oil viscosity changes with the variation of temperature, which significantly affects the HPS damping characteristic [

Geometrical parameters and operating conditions have been proven to have notable effect on the HPS nonlinear characteristics. For example, the diameter of the damping valve has active effect on damping force [

Based on approaches reported in recent studies, the research cost, time, and complexity can be reduced and the modeling accuracy can be improved by combining statistical theory with computer simulation [

Previous studies on the mathematical modeling of the HPS nonlinear characteristics have been mostly based on mathematical derivation and experimental verification [

In this study, a novel method for HPS modeling is proposed. In particular, the principal motivation for this study is to propose a novel modeling method for HPS, which combines CFD simulations with the approximate modeling, and can serve as guidance in designing and controlling HPSs. A flowchart of the overall modeling method is illustrated in Figure

Flowchart of the overall mathematical modeling method of HPS.

This paper is assembled as follows: the structure, working principle, and mathematical modeling method of the HPS under investigation are introduced in Section

The structure diagram of an HPS with single chamber is illustrated in Figure

Structure diagram of the HPS.

The parameters of the HPS are presented in Table

Nomenclature of the HPS parameters.

Parameter | Description | Value |
---|---|---|

Area of the suspension piston | ||

Area of the suspension piston rod | ||

Density of the hydraulic oil | ||

Diameter of the check valve | ||

Diameter of the damping valve | ||

Area of the check valve | ||

Area of the damping valve | ||

Flow rate coefficient |

In general, nitrogen is regarded as an ideal gas in the calculations, and its state change process is described by the polytropic state equation:

The compression force in the cylinder can be calculated as follows:

According to the polytropic state equation, the instantaneous volume and pressure of the nitrogen at any time can be calculated by the following equation:

Then, the stiffness of the hydropneumatic suspension can be calculated as follows:

Consequently, the stiffness of the single-chamber HPS can be obtained as follows:

When the suspension cylinder and the piston rod move up and down relative to each other, the oil flow rate between cavity (I) and cavity (II) can be calculated as follows:

When the hydraulic oil flows through the damping valve, the flow is purely turbulent. Thus, it can be assumed to satisfy the flow formula of thin-walled orifice, which can be written as follows:

The pressure drop

When the Reynolds number

According to Pascal's law, the damping force of the HPS can be calculated by the following equation:

The traditional empirical formula does not take into consideration the effect of the HPS structure and working conditions on stiffness and damping force. However, flow characteristics of the fluid in the HPS, such as pressure and velocity, at different time points can be obtained through CFD simulation, which can be used to calculate the HPS damping and stiffness forces and analyze the nonlinear characteristics. Considering that the HPS under investigation is a single-chamber suspension with gas-oil emulsion and that the hydraulic oil flows freely and the nitrogen state changes with stroke, the changes at the interface between nitrogen and oil need to be tracked. Therefore, the VOF method was employed for the simulations in this study.

The VOF method can simulate the flow and interaction of two or more immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each fluid throughout the domain. The VOF formulation relies on the fact that the different fluids (or phases) are not interpenetrating. For each additional phase that is added to the model, its volume fraction is introduced in the computational cell. If the volume fraction of the ^{th} fluid in the cell is denoted as _{q}, then the following three conditions are possible:

The governing equations of the VOF method include the volume fraction equation, the additional scalar equations, and the continuum surface force model.

The tracking of the interface between the phases is accomplished by solving a continuity equation for the volume fraction of one (or more) of the phases. For the

The volume fraction equation is not solved for the primary phase; the primary-phase volume fraction is computed based on the following constraint:

The density and viscosity of the mixed fluid in the unit are calculated as follows:

The surface tension force is generated by the gravitational force between the molecules in the fluid. It acts only on the surface and minimizes the surface free energy by reducing the interface area. In CFD analysis, the continuum surface force (CSF) model proposed by Brackbill has been used to implement the effect of surface tension force. In this model, the pressure drop across the surface depends upon the surface tension coefficient

The surface tension force can be expressed in terms of the pressure jump across the surface. Using the divergence theorem, the force at the surface can be expressed as a volume force as follows:

If only two phases are present in a cell, the above equation can be simplified as follows:

The volume and pressure of cavity (I) and cavity (II) change dynamically during the compression and extension strokes. In this process, nitrogen dissolves partially in the oil. Due to its low density, the intermolecular distance between nitrogen molecules is relatively larger than that of hydraulic oil molecules, and hence, the nitrogen dissolved in the oil will lead to a decrease in hydraulic oil density and a slight increase in volume.

The dissolution rate of nitrogen in hydraulic oil is defined as the volume of dissolved nitrogen as a percentage of the initial nitrogen volume in HPS. It can be shown as follows:

The density of the gas-oil mixture

Based on the actual HPS structure of a specific mining dump truck, the geometry model was designed using UG software. The internal flow domain of the HPS was extracted for the CFD analysis (as shown in Figure

3D model and isolated flow domain of the HPS.

The unstructured grid method was used to mesh the volume of the flow domain. Since the size of the damping and check valves was smaller than that of the whole model, the mesh at these regions was finer. Finally, the total number of elements in the model was more than 1.6 million. Figure

Cross-section of the generated mesh of the model.

The ANSYS Fluent software was used to perform the transient calculation of the flow in the HPS, and the parameters set to the designed VOF model are listed in Table

VOF model parameters.

VOF model properties | Settings |
---|---|

Primary/secondary phase | Nitrogen gas/oil |

Viscous model | RNG |

Gas state | Ideal gas |

Oil compressibility | Incompressible Newtonian fluid |

Precharge nitrogen pressure ( | 6.276 |

Precharge nitrogen height ( | 100 |

Temperature ( | 300 |

Oil dynamic viscosity ( | 0.1059 |

Oil density ( | 994 |

The dynamic mesh method was used to simulate the relative movement between cylinder and piston rod. The moving boundaries were the cavity (I) upper wall and the cavity (II) bottom wall, as shown in Figure

(a) The defined moving boundaries; (b) plot of boundary velocity versus time in compression stroke; (c) plot of boundary velocity versus time in extension stroke.

The calculation settings for the transient simulation are listed in Table

Calculation settings for the transient simulation.

Solving method | Settings |
---|---|

Solver type | Pressure-based |

Time | Transient |

Near-wall treatment | Standard wall function |

Dynamic mesh method | Smoothing, remeshing |

Pressure-velocity coupling scheme | PISO |

Pressure spatial discretization | Presto! |

Density spatial discretization | First-order upwind |

Volume fraction spatial discretization | Georeconstruct |

Transient formulation | First-order implicit |

Volume changes of oil and nitrogen during the compression stroke at (a)

The internal flow field distribution characteristics of the HPS can be obtained based on the simulation data. The contours of velocity distribution at a cross-section of the damping valve at different time points during the compression stroke are illustrated in Figures

Velocity distribution contours at a cross-section of the damping valve in compression stroke at (a)

Figure

Pressure distribution contours at a cross-section of the damping valve in compression stroke at (a)

The volume ratios of oil and nitrogen in the HPS at representative time points (

Volume changes of oil and nitrogen during the extension stroke at (a)

The velocity distribution contours of flow at a cross-section of the damping valve at

Velocity distribution contours at a cross-section of the damping valve in extension stroke at (a)

The pressure contours at a cross-section of the damping valve in extension stroke at

Pressure distribution contours at a cross-section of the damping valve in extension stroke at (a)

Through the above simulation analysis, the transient pressure distribution in the HPS under specific operating conditions can be determined. Consequently, the transient pressures in cavity (I) and cavity (II) can be obtained to calculate the stiffness and damping of the HPS.

If the friction between cylinder and piston rod can be ignored, the output force of the HPS under external excitation can be calculated as follows:

Since the pressure difference

The output force of the HPS is divided into two parts, stiffness force

Consequently, as long as the transient pressures of cavity (I) and cavity (II) are obtained, the stiffness and damping force of the HPS can be accurately calculated.

The specific implementation scheme of the mathematical modeling process for HPS is presented in Figure

Flowchart of the HPS modeling process.

The surrogate modeling method is an analysis method used in the field of multidisciplinary design optimization (MDO) founded in recent years. It is a highly efficient method for determining the functional or interactional relationship between the input and output of a studied system. In the case of HPS design, the DOE method can be employed to design a large number of simulation sample points of several different factors that affect the HPS nonlinear characteristics. Subsequently, according to the working conditions of these sample points, several groups of CFD simulations can be performed to obtain the HPS damping and stiffness forces under various working conditions. Finally, the simulation data can be fitted by surrogate models, which can build an approximate function relationship between HPS nonlinear characteristics and their influence factors.

The Kriging model, which was employed in this study, is an interpolation method developed in the field of spatial statistics and geostatistics. It predicts the distribution of function values at unknown points instead of the function values themselves. From the distribution of the function values, both function values and their uncertainty at new points can be estimated. Simpson et al. [

The optimal Latin hypercube experiment design was used to provide a uniform distribution of the experimental sample points in the experiment space. In this study, the effects of oil temperature (

L-HM46 hydraulic oil viscosity variation with temperature.

The variation range of each factor was determined empirically (Table

Variation ranges of the four different factors.

Factors | Variation range |
---|---|

Temperature ( | 25∼65 |

Oil viscosity ( | 0.0159∼0.1252 |

Gas dissolution rate (%) | 0∼10 |

Vibration speed ( | −1.0∼1.0 |

Simulation results.

1.0 | 8.95 | 52 | 0.0258 | 5.91 | 29.3 | −363043 | 536437 |

1.0 | 4.95 | 33 | 0.0686 | 5.92 | 30.9 | −387672 | 537618 |

0.9 | 3.68 | 54 | 0.0239 | 5.96 | 25.4 | −300640 | 541068 |

0.9 | 5.37 | 50 | 0.0288 | 5.96 | 25.3 | −300514 | 540714 |

0.8 | 5.26 | 38 | 0.0507 | 5.99 | 21.3 | −237626 | 543828 |

0.8 | 7.89 | 63 | 0.0179 | 5.98 | 21.6 | −242167 | 543193 |

0.7 | 0.53 | 42 | 0.0408 | 6.03 | 17.9 | −183556 | 547769 |

0.7 | 8.32 | 42 | 0.0408 | 6.01 | 17.7 | −181170 | 546126 |

0.6 | 2.11 | 25 | 0.1252 | 6.07 | 15.8 | −150268 | 550820 |

0.5 | 6.32 | 65 | 0.0159 | 6.09 | 12.2 | −94817 | 552954 |

0.4 | 8.42 | 29 | 0.0914 | 6.12 | 10.6 | −68643 | 556096 |

0.3 | 7.89 | 46 | 0.0338 | 6.16 | 8.57 | −37284 | 559609 |

0.2 | 1.05 | 63 | 0.0179 | 6.21 | 7.31 | −17129 | 563423 |

0.1 | 3.16 | 48 | 0.0308 | 6.24 | 6.59 | −5377 | 566556 |

0.1 | 4.53 | 46 | 0.0338 | 6.24 | 6.59 | −5371 | 566519 |

−0.1 | 4.21 | 31 | 0.0795 | 6.31 | 6.24 | 1093 | 573284 |

−0.2 | 0.00 | 36 | 0.0616 | 6.35 | 6.15 | 3046 | 576462 |

−0.2 | 6.63 | 61 | 0.0189 | 6.35 | 6.17 | 2798 | 576961 |

−0.3 | 9.47 | 59 | 0.0199 | 6.40 | 6.04 | 5530 | 580675 |

−0.3 | 7.47 | 44 | 0.0378 | 6.39 | 6.03 | 5684 | 580557 |

−0.4 | 4.74 | 61 | 0.0189 | 6.43 | 5.83 | 9322 | 583790 |

−0.4 | 3.68 | 25 | 0.1193 | 6.43 | 5.75 | 10473 | 583662 |

−0.5 | 10.0 | 40 | 0.0457 | 6.48 | 5.57 | 14164 | 588575 |

−0.5 | 3.26 | 59 | 0.0199 | 6.47 | 5.56 | 14113 | 587176 |

−0.6 | 5.79 | 44 | 0.0378 | 6.51 | 5.22 | 19931 | 590963 |

−0.7 | 6.84 | 27 | 0.1054 | 6.56 | 4.77 | 27643 | 595303 |

−0.8 | 1.58 | 50 | 0.0288 | 6.58 | 4.36 | 34340 | 597219 |

−0.8 | 8.74 | 57 | 0.0219 | 6.60 | 4.43 | 33657 | 599407 |

−0.9 | 2.63 | 33 | 0.0696 | 6.63 | 3.81 | 43575 | 601623 |

−1.0 | 7.37 | 57 | 0.0219 | 6.68 | 3.34 | 51739 | 606771 |

The simulation results were fitted by the Kriging model and response surface model, respectively. Error analysis on the two models was performed, and the statistical results are shown in Table

Evaluation indicators of the Kriging model.

Evaluation indicator | Error acceptance level | Error value | |
---|---|---|---|

Kriging model | Response surface model | ||

Maximum | 0.3 | 0.095 | 0.098 |

Mean | 0.2 | 0.045 | 0.052 |

Root mean square | 0.2 | 0.057 | 0.048 |

0.9 | 0.97 | 0.95 |

The root-mean-square error can be calculated as follows:

R-square of the model is given by the following equation:

From the comparisons between the evaluation index values and the acceptance levels in Table

In order to compare the prediction accuracy of the Kriging model and response surface model on the stiffness and damping characteristics of HPS, additional simulations are required. Therefore, based on the variation ranges of the four factors, 10 groups of sample points were randomly generated, and the validity of the model was verified by comparing the simulated values with the predicted values. Tables

Damping force validation.

No. | Simulation ( | Prediction | Relative error (%) | ||||||
---|---|---|---|---|---|---|---|---|---|

Kriging model ( | Response surface model ( | Kriging model | Response surface model | ||||||

1 | −1 | 0.027 | 50 | 0.0288 | 52.44 | 51.13 | 53.98 | 2.49 | −2.93 |

2 | −1 | 0.05 | 46 | 0.0338 | 52.24 | 51.44 | 51.19 | 1.54 | 2.02 |

3 | −0.6 | 0.09 | 32 | 0.0408 | 19.79 | 20.30 | 19.06 | −2.54 | 3.71 |

4 | −0.6 | 0.03 | 50 | 0.0288 | 19.93 | 19.42 | 19.32 | 2.56 | 3.08 |

5 | −0.5 | 0.05 | 40 | 0.0457 | 14.33 | 14.41 | 14.15 | −0.54 | 1.31 |

6 | −0.3 | 0.036 | 52 | 0.0258 | 5.65 | 5.54 | 4.81 | 1.9 | 14.87 |

7 | 0.6 | 0.05 | 60 | 0.0209 | −136.20 | −135.70 | −138.19 | 0.37 | −1.46 |

8 | 0.8 | 0.07 | 45 | 0.0358 | −240.95 | −237.86 | −237.71 | 1.29 | 1.35 |

9 | 0.9 | 0.05 | 46 | 0.0338 | −301.90 | −304.48 | −298.34 | −0.85 | 1.18 |

10 | 1 | 0.056 | 46 | 0.0338 | −366.77 | −371.06 | −372.66 | −1.17 | −1.61 |

Stiffness force validation.

No. | Simulation ( | Prediction | Relative error (%) | ||||||
---|---|---|---|---|---|---|---|---|---|

Kriging model ( | Response surface model ( | Kriging model | Response surface model | ||||||

1 | −1 | 0.027 | 50 | 0.0288 | 605.32 | 604.85 | 605.96 | 0.08 | −0.11 |

2 | −1 | 0.05 | 46 | 0.0338 | 587.35 | 589.83 | 609.08 | −0.42 | −0.55 |

3 | −0.6 | 0.09 | 32 | 0.0408 | 591.80 | 591.98 | 590.87 | −0.03 | 0.16 |

4 | −0.6 | 0.03 | 50 | 0.0288 | 590.73 | 592.89 | 593.69 | −0.36 | −0.50 |

5 | −0.5 | 0.05 | 40 | 0.0457 | 587.35 | 589.83 | 589.96 | −0.42 | −0.44 |

6 | −0.3 | 0.036 | 52 | 0.0258 | 580.15 | 580.54 | 580.96 | −0.07 | −0.14 |

7 | 0.6 | 0.05 | 60 | 0.0209 | 550.15 | 547.65 | 556.30 | 0.46 | −1.12 |

8 | 0.8 | 0.07 | 45 | 0.0358 | 543.20 | 539.25 | 538.64 | 0.73 | 0.84 |

9 | 0.9 | 0.05 | 46 | 0.0338 | 540.71 | 537.05 | 536.05 | 0.68 | 0.86 |

10 | 1 | 0.056 | 46 | 0.0338 | 537.61 | 535.46 | 539.87 | 0.4 | −0.42 |

The Kriging model cannot be shown as an explicit formulation. The surrogate models of HPS nonlinear damping and stiffness are shown in Figures

Surrogate models of the relationship between (a) damping force, oil viscosity, and gas dissolution rate; (b) damping force, temperature, and oil viscosity; (c) damping force, suspension vibration speed, and temperature; and (d) damping force, suspension vibration speed, and oil viscosity.

Surrogate models of the relationship between (a) stiffness force, gas dissolution rate, and oil viscosity; (b) stiffness force, suspension vibration speed, and dissolution rate; (c) stiffness force, suspension vibration speed, and temperature; and (d) stiffness force, suspension vibration speed, and oil viscosity.

Figure _{c} on the curved surface is nearly parallel, independent of the value of T. This means that the interaction between

Figure _{k}-_{k}-

By postprocessing the surrogate model, the response sensitivity to factors can be analyzed. The Pareto plots for the damping and stiffness of the HPS are shown in Figures

Pareto plots of HPS: (a) damping and (b) stiffness characteristics.

In order to verify the accuracy of the Kriging model, an experiment using a mining dump truck and a 11-DOF mathematical model are proposed. The accuracy of the traditional HPS model and the proposed HPS model was compared by analyzing the RMS acceleration of the vehicle wheel center and measuring the point of driver’s feet.

The driving experiment using the mining dump truck was performed in an open pit mine in western China (Figure

Mining dump truck and measurement instrumentations.

Experimental devices.

Experimental device | Assignment |
---|---|

Battery and invertor | Supply energy for the device |

Accelerometer, | Measuring the acceleration |

Amplifier | Amplify the measured signal |

Recorder, MR30 C | Record the signal |

Data analysis device and software, SD380 | Analyze the signal |

Measurement points of experiment.

The experimental results concerning the suspension upper point at a speed of

Plots of the measured suspension upper point acceleration versus time (at 30 km/h) for the (a) front left suspension; (b) front right suspension; (c) rear left suspension; and (d) rear right suspension.

In order to verify the proposed HPS model, an 11-DOF mathematical model of the mining dump truck was built (Figure

Diagram of the 11-DOF truck model, where

According to Newton’s Law, the dynamic mathematical equation of vertical axles, the pitch, and roll can be written as follows:

Considering the kinetic coordination of the four suspensions, the suspension deflection equation can be calculated as follows:

The plots of RMS acceleration of the four suspensions versus vehicle velocity are illustrated in Figure

Plots of RMS acceleration versus vehicle velocity for the four suspensions. Comparison between the experimental, traditional models, and proposed model results concerning the upper point of the (a) front left suspension; (b) front right suspension; (c) rear left suspension; and (d) rear right suspension.

The aim of this study was to propose a high-precision modeling method of the nonlinear characteristics of HPS in which the effect of multiple influence factors can be considered. The oil temperature, oil viscosity, gas dissolution rate, and suspension vibration speed were investigated as the influencing factors of the nonlinear characteristics of HPS. The findings of this study are summarized as follows:

The mathematical modeling method for HPS proposed in this paper, which combines CFD simulation with the approximate model method, provided a highly efficient and accurate way to develop a simple and direct mathematical model describing and predicting the relationship between HPS nonlinear characteristics and the influencing factors regardless of structural changes in HPSs, which will substantially reduce the experimental costs and speed up the development process. This has very important guiding significance for the analysis and development of HPSs.

The Kriging model demonstrated excellent performance for approximating the mathematical relationship between HPS nonlinear characteristics and influencing factors. The results suggested that it can provide a good prediction accuracy for the damping and stiffness forces of HPS with the prediction errors not exceeding 5%.

However, this modeling method cannot be expressed explicitly, and the effect of using different gas state equations on the accuracy of the transient calculations is not taken into consideration. In future research work, where more factors will be considered, the applicability of this method needs to be verified. In addition, further investigation is worth to be done to determine how such a method can be applied to enhance active suspension control and autonomous driving, in order to improve driving comfort.

All data included in this study are available upon request by contact with the corresponding author.

The authors declare that they have no conflicts of interest.

Special thanks to the National Natural Science Foundation of China (51705035) and the Natural Science Foundation of Hunan Province (2017JJ3336). Contribution was also supported by XEMC.