Keywords

1 Introduction

Suspension design is crucial for ensuring the safety of personnel and extending the lifespan of vehicles, making it a key aspect of vehicle design. Accurately analyzing the dynamic characteristics of the suspension using precise mathematical models is highly significant. Currently, hydro-pneumatic suspension cylinders with exceptional elasticity and damping are utilized for mine vehicle suspensions, specifically designed to tackle complex road conditions. A lot of basic and systematic researches have been done on hydro-pneumatic suspension, and the formation mechanism and modeling method of nonlinear stiffness and damping characteristics have been analyzed in detail [1, 2]. The elastic force is modeled by the variable state equation of gas [3, 4], while the damping force is modeled by thin-walled orifice theory, whether it is 1/4 vehicle suspension simulation [5, 6], complete vehicle suspension vibration reduction optimization [7, 8], multi-axle vehicle suspension modeling [9], or active suspension adjustment research [10, 11]. However, it has been observed that the performance of these models is less than ideal in both the frequency domain and time domain [7, 12]. Recently, the use of CFD technology has emerged to obtain nonparametric results and fit them to modify damping characteristics [13, 14]. Although this method provides results, it is not conducive to the subsequent optimization of suspension cylinder design. Therefore, it is necessary to further investigate the dynamic model of the suspension cylinder. In the past, only the flow in the orifice was usually considered to be in a turbulent state. Still, the suspension cylinder was always in a frequent up-and-down bumping state during the driving process of the vehicle. The flow field in the orifice constantly developed from laminar flow to transition and then entered the turbulent zone. The changing direction is reciprocated, so the actual situation should include laminar flow and transition zone. Especially according to Nikolaz's graph or Moody diagram [15], the resistance along the transition zone has a violent change process, including chaos in this area, which was narrow and ignored in the past. However, considering that the sharp change of flow field in this area will affect the dynamic characteristics in the frequency domain, it is necessary to consider the whole change process of flow in the damping orifice and one-way valve from static to complete development of turbulence. In addition, because the ratio of the length and diameter of the damping hole of the suspension cylinder is not thin-walled, and it is actually in the range of short-hole flow theory, a more realistic model should be constructed according to the short-hole flow theory. This paper aims to reconstruct the damping force using the piecewise function expression method of the short-hole flow theory. Additionally, two theoretical models are adopted to represent the turbulent region: the Blasius formula, recommended by the short-hole flow theory, which offers high accuracy and broad applicability, and the traditional thin-walled orifice theoretical model in the time and frequency domains are compared, and the accuracy of several theories, especially their performance in the frequency domain, is analyzed and discussed.

2 Damping Force Mathematical Model

The structural diagram of the conventional suspension cylinder is shown in Fig. 1.

Fig. 1
A schematic representation of the cross section of oil pneumatic spring. It is composed of a piston rod, chamber 2, chamber 1, a one-way valve, a damping hole, and a cylinder.

Structural diagram of oil pneumatic spring

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The fluid resistance of the damping hole and one-way valve is generated by the pressure difference between chamber I and chamber II, and the nonlinear damping Fc is:

$$F_{C} = {\text{A}}\Delta P$$
(1)

In the formula: \(\Delta P\) is the pressure difference between cavity I and cavity II; A is the radial area of the buffer chamber II. The main differences between the two mathematical models of nonlinear damping forces are as follows.

2.1 Thin-Walled Orifice Theory

The relationship between the pressure difference between chamber I and chamber II and the flow rate Q through the damping hole and one-way valve is:

$${\text{Q}} = {\text{C}}_{{\text{d}}} A_{1} \sqrt {\frac{2\Delta P}{\rho }}$$
(2)

In the formula, Cd is the flow coefficient; ρ is the density of the oil; A1 is the total flow area through the damping hole and one-way valve.

Bringing Formula (2) into (1), while considering that the flow rate Q is the flow rate from cavity I to cavity II, it can be obtained by multiplying the velocity difference between the upper and lower support points of the suspension cylinder by A. Assuming that the flow rate in the damping hole and the one-way valve is the same, we obtain the relationship between the final suspension cylinder damping force and the velocity difference between the upper and lower support points:

$${\text{F}}_{C} = \frac{{\rho A^{3} \upsilon^{2} {\text{sign}}\upsilon }}{{2\left\{ {} \right.{\text{nC}}_{{\text{d}}} \left. {[A_{{\text{d}}} + A_{{\text{v}}} (0.5 + 0.5{\text{sign}}\upsilon )]} \right\}^{2} }}$$
(3)

In the formula: n is the number of damping holes and one-way valves; Ad is the cross-sectional area of the damping hole; Av is the effective flow area of the one-way valve; \(\upsilon\) is the speed of the cylinder barrel relative to the piston; Sign \(\upsilon\) is the sign function of velocity, which is compressed as positive and restored as negative based on the direction of velocity.

2.2 Short-Hole Flow Theory

According to formulas 8.32 in [15], the head losses of cavity I and cavity II are composed of three parts: inlet head loss, short-hole overflow head loss, and outlet head loss. The relationship between pressure difference and flow velocity through damping holes and one-way valves is as follows:

$$\Delta P = (\zeta_{inlet} + \zeta_{outlet} + \zeta_{channel} )\frac{{\rho \upsilon_{H}^{2} }}{2}$$
(4)

In the formula, \({\upzeta }_{{\text{inlet}}}\), \({\upzeta }_{{\text{outlet}}}\) and \({\upzeta }_{{\text{channel}}}\) are the local head loss coefficients along the inlet, outlet, and short hole respectively; \(\upsilon_{H}\) is the flow rate inside the hole. The inlet resistance coefficient \(\zeta_{inlet}\) can be based on the data obtained from the Weissbach experiment, and considering the shrinkage situation, the value is 0.5 when a container with a large cross-section flows into the pipeline. The outlet resistance coefficient \(\zeta_{outlet}\) is based on the Borda formula and is considered when the fluid flows into a container with a large cross-section under submerged conditions. Therefore, the local head loss is taken as 1. The local head loss coefficient along the short hole \({\upzeta }_{{\text{channel}}}\) is calculated in sections based on the changes in the flow field inside the small hole.

2.3 Damping Force in Laminar Flow Zone

The laminar flow zone is the initial stage of the flow field, with Re ranging from 0 to 2300. The local head loss coefficient in this range is inversely proportional to Re:

$${\upzeta }_{{{\text{channel}}}} = \frac{{{64}}}{{R{\text{e}}}}\frac{L}{{\text{d}}}$$
(5)

Here \(L\) is the wall thickness of the small hole, and d is the diameter of the small hole. \(R{\text{e}}\) is the Reynolds number, indicating the turbulent state of flow. When this value is less than 2300, it is laminar flow, which is related to velocity, flow field size, and viscosity:

$${\text{Re}} = \frac{{\upsilon_{H} d_{{}} }}{\nu }$$
(6)

In the formula: \(\nu\) is the kinematic viscosity of the fluid oil in the suspension cylinder; \(d_{H}\) is the diameter of the small hole.

Formula (6) is brought into (4) to get the relationship between the damping force and the flow velocity in the small hole:

$$F_{C} = \left( {1.5 + \frac{{{64}}}{{R{\text{e}}}}\frac{{\text{L}}}{{\text{d}}}} \right)\frac{{\rho \upsilon_{{\text{H}}}^{2} }}{2}A$$
(7)

Similar to the form of the Formula (3), a formula for constructing a compression state that can be applied to both the open and closed recovery states of a one-way valve is constructed. The relationship between the flow in the small hole \(\upsilon_{{\text{H}}}\) and the velocity difference \(\upsilon\) between the upper and lower support points of the suspension cylinder during the recovery and compression processes is:

$$\upsilon_{{\text{H}}} \left( \upsilon \right) = \frac{{A^{{}} \upsilon }}{{{\text{nC}}_{{\text{d}}} \left[ {{\text{A}}_{{\text{d}}} {\text{ + A}}_{{\text{v}}} \left( {{0}{\text{.5 + 0}}{\text{.5sign}}\upsilon } \right)} \right]}}$$
(8)

The relationship between the damping force and the speed difference between the upper and lower support points of the suspension cylinder is obtained by bringing formula (8) into (6) and then into (7) as follows:

$$F_{C} = \left( {1.5 + \frac{{{64}v{\text{L}}}}{{\upsilon_{H} \left( \upsilon \right){\text{d}}^{2} }}} \right)\frac{{\upsilon_{H}^{{^{2} }} \left( \upsilon \right)}}{2}\rho A$$
(9)

2.4 Damping Force in the Transition Zone

It is generally believed that the transition zone, due to its narrow range and limited practical significance, is not discussed. However, the transition zone produces a significant change in frictional coefficient along the way under a small Reynolds number change, which inevitably leads to an increase in fluid noise. Therefore, it is necessary to analyze its impact. The range of Re in the transition zone is between 2300 and 3000, and it can be seen from the Nicholas curve that there is a clear chaotic state. Therefore, there has been little research in the past. According to the empirical formula provided by [15], the relationship between the local head loss coefficient \({\upzeta }_{{\text{channel}}}\) and Re is as follows:

$${\upzeta }_{{{\text{channel}}}} = 0.0025R_{e}^{1/3}$$
(10)

The relationship between damping force and flow velocity in the small hole:

$$F_{C} = \left( {1.5 + 0.0025R_{e}^{1/3} \frac{{\text{L}}}{{\text{d}}}} \right)\frac{{\rho \upsilon_{{\text{H}}}^{2} }}{2}A$$
(11)

Similarly, Formulas (8) and (6) are introduced into (11) to obtain the relationship between the damping force and the velocity difference \(\upsilon\) between the upper and lower support points of the suspension cylinder. A formula is constructed for the compression state with both the open and closed one-way valves, and the relationship between the damping force and the velocity \(\upsilon\) between the upper and lower support points of the suspension cylinder becomes:

$$F_{C} = \left( {1.5 + { 0}{\text{.0025}}\left( {\frac{{\upsilon_{H} (\upsilon ){\text{d}}}}{\nu }} \right)^{\frac{1}{3}} \frac{{\text{L}}}{{{\text{d}}^{{}} }}} \right)\frac{{\upsilon_{H}^{{^{2} }} (\upsilon )\rho_{{}} }}{2}A$$
(12)

2.5 Damping Force in the Turbulent Zone

The damping force in the turbulent zone adopts two models, namely the damping force formula for thin-walled orifice (3) and the Blasius model in the short-hole flow theory. Brasius is a widely applied and mature empirical formula provided by the short-hole flow theory for turbulent regions. The calculation of the resistance loss coefficient \({\upzeta }_{{\text{channel}}}\) along the path in the range of 3000 < Re < 10e5 using this formula has high accuracy. The relationship between the local head loss coefficient \({\upzeta }_{{\text{channel}}}\) and Re is as follows:

$$\zeta_{channel} = \frac{0.3164}{{{\text{Re}}^{0.25} }}\frac{{L_{{}} }}{{d_{{}} }}$$
(13)

Similarly, Formulas (8) and (6) are taken into (13) and then taken into (4) to obtain the relationship between the damping force with both compression and recovery states and the velocity difference \(\upsilon\) between the upper and lower support points of the suspension cylinder:

$$F_{C} = \left( {1.5 + { 0}{\text{.3164}}\left( {\frac{\nu }{{\upsilon_{H} (\upsilon ){\text{d}}}}} \right)^{\frac{1}{4}} \frac{{\text{L}}}{{{\text{d}}^{{}} }}} \right)\frac{{\upsilon_{H}^{{^{2} }} (\upsilon )\rho_{{}} }}{2}A$$
(14)

2.6 Summary of Piecewise Function Damping Force Models

The damping force of the suspended cylinder is expressed using a piecewise function based on the short-hole flow theory, which is laminar flow in the early development stage of Re ≤ 2300, the transition of 2300 < Re < 3000, and turbulence of 3000 ≤ Re. According to Sect. 2.3 and 2.4, the laminar damping force and transition zone damping force are calculated using formula (3) and Brasius' formula for the turbulent zone:

$$\left\{ \begin{gathered} - - - - - - - - - - - - - - - - \hfill \\ {\text{flow progress expressed by piecewise:}} \hfill \\ R{\text{e}} < {\text{2300 laminar flow zone:}} \, \hfill \\ {\text{formula }}\left( 9 \right) \hfill \\ {2300} < R{\text{e}} < {3}0{\text{00 transient zone:}} \hfill \\ {\text{formula }}\left( {12} \right) \hfill \\ \, R{\text{e}} > {3}0{\text{00 turbulence zone with 2 model:}} \hfill \\ {\text{formula }}\left( {14} \right) \hfill \\ {\text{formula }}\left( 3 \right) \hfill \\ - - - - - - - - - - - - - - - - \hfill \\ {\text{the entire flow progress:}} \hfill \\ {\text{thin-walled model }}\left( 3 \right) \hfill \\ \end{gathered} \right.$$
(15)

So, two piecewise function models are established here, with different formulas used in the turbulent zone. Finally, the calculation results are compared with those of the traditional thin-walled orifice theory model (3) used throughout the entire flow field, and the influence of different damping force models on the results is observed.

2.6.1 Dynamics Model of Suspension Cylinder

The lower fulcrum of the suspension cylinder is excited by external input shock or road displacement, and the vibration reduction of the upper fulcrum is achieved through the elasticity of the suspension cylinder's air chamber and fluid damping. The dynamic model of the upper fulcrum based on Newton's third law of motion is:

$${\text{m}}\dot{\upsilon } = F_{C} + F_{gas}$$
(16)

In the formula, m represents the mass on the spring, \(\dot{\upsilon }\) represents the acceleration of the support point on the suspension cylinder, \(F_{{\text{c}}}\) represents the damping force in Formula (15), and \(F_{gas}\) represents the elastic force derived from the gas variable state equation. The weight on the spring is reflected by the gas compression amount in the elastic equation in the static equilibrium state.

3 Simulation Comparison of Three Kinds of Damping Force Mathematical Models

The common suspension simulation generally uses pulse input to verify the impact characteristics or a random road surface model constructed by the power spectral density function for suspension dynamics simulation in the absence of road test data [16]. A more accurate method is to collect the suspension vibration and driver's body acceleration when the vehicle is driving on real mining roads, using the standards for evaluating vehicle smoothness specified in ISO2631 [14], grading to evaluate the vehicle's suspension damping capability. This method has been widely promoted and applied in the automotive industry.

Here, the lower fulcrum of the suspension cylinder under real road conditions is used as the model input, and the sampling results of the upper fulcrum are compared with the simulation calculation results to verify the accuracy and difference between the two damping force models. The data collection of real road conditions adopts the method of installing an acceleration sensor at the top of the front axle, which is rigidly connected to the front suspension cylinder barrel, to obtain the acceleration of the lower fulcrum of the oil and gas suspension cylinder. An acceleration sensor is installed at the top of the piston rod to obtain the acceleration of the upper fulcrum of the oil and gas suspension cylinder, with a sampling period of 5e-4 s. As the input of Formula (15), the motion of the lower fulcrum can obtain the acceleration data of the upper fulcrum, which can be compared with the actual sampling results. Figure 2 shows the comparison between the calculation results of three models in the time domain and the real fulcrum values. The horizontal axis represents the time, and a typical time-domain comparison chart from 1.55 to 1.95 s is taken to display the simulation results.

Fig. 2
A graph plots a versus t. 3 sinusoidal functions for thin walled orifice and piecewise function formulas 3 and 14 are plotted between negative 1.5 and 2.5. The line for real fulcrum data is plotted along the sinusoidal curves with some fluctuations between negative 4 and 4.

Comparison of calculation results and real values between two models in the time domain

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Firstly, from the overall trend, it can be seen that all three models can follow the excitation signal to make a response that is close to the real one. However, there is a lag delay phenomenon in the wave peaks of all three models, and the thin-walled orifice theory model is more evident at individual moments, such as at 1.7 s. In addition, it can be seen that the two piecewise functions models have better tracking performance, and most of the time remain within the fluctuation range of the true value. At 1.75–1.8 s, it can also better simulate small fluctuations in real vibration. The two piecewise functions do not exhibit particularly significant differences in the time domain. However, all three models exist: they cannot be simulated on finer vibrations, i.e. losing the high-frequency part; Calculate the situation where the vibration amplitude is lower than the true value, as shown in the peaks and valleys in Fig. 2.

Figure 3 shows the comparison between the three theoretical models in the frequency domain and the actual sampling results. The frequency domain is obtained by adding a recovery coefficient of 1.633 and a power recovery coefficient of 2 through Fourier transform after the hamming window. It can be seen from the figure that both models can reflect the low-frequency vibration, but the attenuation is fast. After about 15 Hz, the calculated results show a significant attenuation of amplitude, which cannot simulate the real situation. After 20 Hz, the real frequency cannot be simulated.

Fig. 3
A graph plots a versus f. 4 curves for real fulcrum data, thin walled orifice, and piecewise function formulas 3 and 14 begin at (0, 0) and remain stable along the x axis from 0 to 60 hertz with several minimum and maximum peaks between 0 and 0.8 meters per second square.

Frequency domain comparison between the calculated results of three models and the true values

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Figure 4 is an enlargement of the low-frequency region of Fig. 3, showing that the calculated values of the three models at the maximum vibration value of the suspended cylinder are all greater than the real upper fulcrum values. The results of the thin-walled orifice theory model are more detached from the actual situation, and the two piecewise function models are relatively close to the actual values. The attenuation degree of the two piecewise function models above 6 Hz is also better than that of the thin-walled orifice theory, and they can still calculate a certain acceleration amplitude. The performance of the two piecewise functions in the frequency domain is almost identical visually, with no significant differences.

Fig. 4
A graph plots a versus f. 4 curves for real fulcrum data, thin walled orifice, and piecewise function formulas 3 and 14 begin at (0, 0) and remain stable along the x axis from 0 to 8 hertz with several minimum and maximum peaks between 0 and 0.8 meters per second square.

Comparison of three models and real fulcrum values in the low-frequency range

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To quantitatively evaluate the accuracy of the three models, the root mean square error (RMSE) and mean absolute error (MAE) of the corresponding frequency values in the frequency domain are introduced to measure the degree to which the two models deviate from the true value. MAE (Mean Absolute Error) is the average absolute value of the error between the calculated value and the true value. The formula is:

$$MAE = \frac{1}{{\text{n}}}\sum\limits_{{{\text{i}} = 1}}^{{\text{n}}} {\left| {\left. {{\hat{\text{y}}}_{{\text{i}}} - {\text{y}}_{{\text{i}}} } \right|} \right.}$$

The Root Mean Square Error (RMSE) is concentrated on the order of magnitude for comparison, such as RMSE = 10. It can be considered that the regression effect has an average difference of 10 compared to the true value.

$$RMSE = \sqrt[{}]{{\frac{1}{{\text{n}}}\sum\limits_{{{\text{i}} = 1}}^{{\text{n}}} {\left( {{\hat{\text{y}}}_{{\text{i}}} - {\text{y}}_{{\text{i}}} } \right)^{2} } }}$$

When the calculated value completely matches the actual value, it is equal to 0, which is a perfect model; the larger the error, the greater the value. Two indicators are used to describe the error between calculated values and true values. The difference is that if the dispersion of the error is high and the maximum deviation value is large, RMSE will amplify compared to MAE, which can better reflect the degree of deviation. In Table 1, the input parameters and output performance of the three models are listed as “×” indicating that the parameter is not required.

Table 1 Comparison of the advantages and disadvantages of three damping force models
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In Table 1, the performance of three models in the time domain MAE and RMSE under a duration of 20 s was compared, while the performance in the frequency domain was compared within the range of 0–100 Hz. From Table 1, it can be seen that the damping force constructed using piecewise functions has better accuracy than the thin-walled orifice theory model. The difference between the two piecewise functions is not significant. From the perspective of formula complexity, Formula (3) can be selected as the piecewise function form of the turbulent zone model to construct the damping force.

4 Conclusion

This article modifies the dynamic characteristics of the hydraulic and pneumatic suspension of mining vehicles, especially the damping force model, according to the actual structural characteristics of the suspension cylinder. The short-hole flow theory model is used to consider the entire process of the flow field development in the small hole, which includes the initial laminar flow, the transient transition in the middle, and the turbulence in the later stage to reconstruct the piecewise damping force model. By comparing the system response under real road excitation with the traditional thin-walled orifice theoretical model, the average absolute error MEA root mean square error RMSE was compared in both time and frequency domains. Compared with thin-walled orifices, the piecewise short-hole flow model improved the accuracy in the time domain by 24–25%, frequency domain accuracy MAE by about 15.8%, and RMSE by 37.4%. It is verified that the damping force of the piecewise short-hole flow model is closer to the real situation in the frequency and time domains. From the perspective of simplicity and ease of use, using the piecewise short-hole flow model using Formula (3) in the turbulent zone is recommended. Its time-domain and frequency-domain MEA and RMSE are close to the Formula (14) but with low complexity.

Through the results in the time domain and frequency domain, it can be found that the performance of the three damping force models in the middle and high-frequency parts, except the low frequency has an obvious suppression phenomenon, which may be related to three chaotic phenomena: (1) Chaotic phenomenon in the transition zone shown by Nikolaz graph or Moody graph; (2) Chaotic motion of steel ball in flow field in one-way valve [14]; (3) The oil sloshing in the oil–gas suspension cylinder causes the internal flow field to be in a chaotic state. The above three situations may be the main reasons that lead to the poor simulation of the dynamic characteristics of suspension cylinders by the existing theoretical models, and further research is needed to find a more accurate mathematical model.