Parametric Design Method of Hydraulic Buffer System for Low-Speed Heavy-Load Trailer

Abstract

To address the problem of high-pressure low-frequency hydraulic impact on the hydraulic buffer system mounted on self-propelled hydraulic trailer, a parametric simulation and design framework is provided for the widespread use of integrated accumulator systems in the low-speed heavy-load vehicle. A mathematical model of the accumulator was established for theoretical analysis, and numerical simulation were conducted on corresponding parameters using AMESim. Finally, a routine method was proposed to achieve feasible arrangement of accumulators. In this study, for the problem of high-pressure impact conditions caused by severe fluctuations, the system performance could be increased by arranging different combinations of accumulators with different parameter configurations. Taking a commercial model of heavy trailer as an actual case, a reasonable parameterized design was carried out for its comprehensive accumulator buffer system, and verification was conducted.

1 Introduction

The low-speed heavy-load vehicle such as self-propelled hydraulic trailer could experience periodic and severe fluctuations in external loads during working operation, causing rapid reversal or stagnation of the fluid flow in the hydraulic buffer system, which results in low-frequency and high-strength hydraulic impacts [1].

As an important component of the hydraulic buffer system, the working principle is to convert excess energy into potential energy for storage based on energy balance. When the hydraulic buffering system needs it, the stored energy can be released [2]. The airbag type accumulator has relatively small inertia, fast response, and the characteristics of oil gas separation, less leakage, and easy maintenance, which make it widely used in the design of hydraulic buffer system [3, 4]. Since WABCO Company proposed the hydropneumatic suspension technology in the 1950s, hydraulic buffer systems based on accumulators have been widely used in low-speed and heavy-load vehicles and construction machinery. For example, various types of mining dump trucks, all terrain cranes, wheeled excavators, and self-propelled hydraulic trailers.

The use of computer simulation technology to study the actual dynamic characteristics of hydraulic system and achieve parameterized design is an important tool for developing modern hydraulic system research technology. For a certain type of system family, including single component design selection and system combination optimization configuration, has become a new idea in the current hydraulic buffer system design [5]. This study takes a type of heavy-load transport trailer as an actual case and provides a parameter simulation path and a combination optimization framework for the design of hydraulic buffer system family.

2 Mathematical Analysis

Taking the airbag type accumulator as the research object, the simplified model [6] of the accumulator is shown in Fig. 1.

Fig. 1

Simplified model of energy accumulator

The inflation gas in the airbag accumulator is nitrogen, which is regarded as an ideal gas and satisfies the equation:

$${p}_{a0}{V}_{a0}^{n}={p}_{a2}{V}_{a2}^{n}=C$$

(1)

where ${p}_{a0}$ (${\text{MPa}}$) is initial pressure and ${p}_{a2}$ (${\text{MPa}}$) is the final pressure, ${V}_{a0}$ (${{\text{m}}}^{3}$) is the initial inflation volume and ${V}_{a2}$ (${{\text{m}}}^{3}$) is the final inflation volume in the accumulator. ${\text{n}}$ is polytropic exponent, which is between 1 and 1.4 in practical gas compression process.

Simplifying airbag type accumulator into a system with only axial motion indicates that the system is a spring damping system. The force equation of the charging chamber of the accumulator is deduced as

$$(p_{b} - p_{a} )A_{a} = k_{e} \frac{{V_{a} }}{{A_{a} }} + c_{e} \frac{1}{{A_{a} }}\frac{{dV_{a} }}{dt}$$

(2)

where ${p}_{b}$ (${\text{MPa}}$) is oil pressure and ${p}_{a}({\text{MPa}})$ is gas pressure in the accumulator. ${A}_{a}$ (${{\text{m}}}^{2}$) is section area of the accumulator. ${k}_{e}$ is stiffness coefficient, ${V}_{a}$ (${\text{MPa}}$) is volume and ${c}_{e}$ is damping coefficient of the gas in the accumulator. ${k}_{e}$ and ${c}_{e}$ are defined [7] by:

$${k}_{e}=\frac{\Delta F}{\Delta x}=\frac{\Delta p\times A}{\Delta V/A}={{A}_{a}}^{2}\frac{dp}{dV}={{A}_{a}}^{2}\frac{n{p}_{a0}}{{V}_{a0}}$$

(3)

$${c}_{e}=8\pi {\mu }_{1}l=8\pi \mu \frac{{V}_{a}}{{A}_{a}}$$

(4)

where ${\mu }_{1}$ is the viscosity coefficient of the inflated gas in the accumulator.

Due to the fact that the liquid stiffness of the hydraulic oil in the accumulator is much greater than the gas stiffness, the elastic modulus of the oil is not considered in the force analysis of the hydraulic oil in the airbag accumulator. The force equation of the oil inlet chamber of the accumulator is deduced as:

$${p}_{1}{A}_{n}-{p}_{a}{A}_{a}=m\frac{{d}^{2}{V}_{a}}{d{t}^{2}}\frac{1}{{A}_{a}}+{B}_{e}\frac{d{V}_{a}}{dt}\frac{1}{{A}_{a}}$$

(5)

where ${p}_{1}$ (${\text{MPa}}$) is inlet pressure of the accumulator. $m$ (${\text{kg}}$) is quality and ${B}_{e}$ is viscous damping coefficient of the hydraulic oil. ${B}_{e}$ has the form:

$$B_{e} = 8\pi \mu_{2} (l_{a} + l_{n} )$$

(6)

where ${\mu }_{2}$ is dynamic viscosity coefficient of hydraulic oil, ${l}_{a}$ (${\text{m}}$) is length of oil inlet and ${l}_{n}$ (${\text{m}}$) is pipe length between hydraulic system and accumulator.

Now assuming ${A}_{n}=k{A}_{a}$ and substituting into Eq. 2 and Eq. 5, the force equation of the accumulator is obtained as:

$$kp_{1} - p_{a} = \frac{1}{{A_{a}^{2} }}\left( {m\frac{{d^{2} V_{a} }}{{dt^{2} }} + B_{e} \frac{{dV_{a} }}{dt} + c_{e} \frac{{dV_{a} }}{dt} + k_{e} V_{a} } \right)$$

(7)

With reference to Eq. 1, there is rewritten as:

$${p}_{a0}{V}_{a0}^{n}={p}_{a}{V}_{a}^{n}$$

(8)

$$\frac{d{V}_{a}}{dt}=-\frac{{V}_{a}}{n{p}_{a}}\frac{d{p}_{a}}{dt}=-\frac{{V}_{a0}}{n{p}_{a0}}\frac{d{p}_{a}}{dt}$$

(9)

Since the elastic modulus of the oil is not taken into account, the change in the gas chamber is equal to the change of the hydraulic oil entering the liquid chamber in the accumulator. The oil flow rate of system $Q$ has the form:

$$Q=-\frac{d{V}_{a}}{dt}=\frac{{V}_{a0}}{n{p}_{a0}}\frac{d{p}_{a}}{dt}$$

(10)

By Laplace transform, Eq. 11 is rewritten as：

$$Q\left(s\right)=\frac{{V}_{a0}}{n{p}_{a0}}(s)$$

(11)

By combining Eqs. 7 and 11, the whole system transfer function $G\left( s \right)$ can be derived as:

$$G\left(s\right)=\frac{{V}_{a}(s)}{{p}_{1}(s)}=\frac{k{V}_{a}^{2}}{m{s}^{2}+\left({B}_{e}+{c}_{e}\right)-(\frac{n{p}_{a0}{A}_{a}^{2}}{{V}_{a}^{2}}-{k}_{e})}$$

(12)

which reduces to:

$$G\left(s\right)=\frac{k{V}_{a}^{2}}{n{p}_{a0}{A}_{a}^{2}-{k}_{e}{V}_{a0}}\cdot \frac{{\omega }_{n}^{2}}{n{s}^{2}+2\xi {\omega }_{n}+{\omega }_{n}^{2}}$$

(13)

where ${\omega }_{n}$ is the natural frequency and $\xi$ is equivalent damping ratio of airbag accumulators, having the form:

$${\omega }_{n}=\sqrt{\frac{n{p}_{a0}{A}_{a}^{2}}{{V}_{a0}m}-\frac{{k}_{e}}{m}}$$

(14)

$$\xi =\frac{{B}_{e}+{c}_{e}}{2\sqrt{\frac{n{p}_{a0}{A}_{a}^{2}m}{{V}_{a0}}-{k}_{e}m}}$$

(15)

3 System Simulation and Analysis

In this paper, the hydraulic buffering system is mainly used for vibration buffering of low-speed heavy-load trailers. In the vibration analysis, vibration generally could be simplified and analyzed. Vehicle vibration can be simplified into models with multiple degrees of freedom. This article adopts a two-degree-of-freedom model. As shown in Fig. 2, the impact of tire elasticity on vibration is more considered than the single-degree-of-freedom model.

Fig. 2

Two-degree-of-freedom model for suspension vibration

Take a certain commercial model of heavy-load trailer as an example shown in Fig. 3. Based on the physical model of the accumulator shown in Fig. 1 and the suspension buffering model shown in Fig. 2, a simulation system [7] is established as shown in Fig. 4. The parameters of each component in the system are shown in Table 1.

Fig. 3

A certain commercial heavy-load trailer as example

Fig. 4

Hydraulic buffer system model based on AMESim

Table 1 Parameters of various components

The impact buffering efficiency of accumulator group is now evaluated. Due to the long-term impact conditions of low-speed heavy-load trailers, the buffering effect is more significantly reflected in the maximum displacement of the trailer body after each impact compared to the decay time of a single step response. Similarly, external impact load from road will also be reflected in the form of impact displacement of the trailer body.

Based on the maximum displacement of the trailer body, under a certain load pressure, the impact displacement rate ${k}_{r}$ and the buffering efficiency $\eta$ are defined by:

$${k}_{r}=\frac{\left|{x}_{max}-{x}_{min}\right|}{\left|{x}_{end}\right|}$$

(16)

$$\eta =\frac{{k}_{r0}-{k}_{r1}}{{k}_{r0}}\times 100\%$$

(17)

where ${x}_{max}$ and ${x}_{min}$ are related to the highest and lowest height of the trailer body in response to impact load. ${x}_{end}$ is the final height of the trailer body. ${k}_{r0}$ is the impact displacement rate without accumulator group. ${k}_{r1}$ is the impact displacement rate with accumulator group. The higher $\eta$, the better the buffering effect of the system compared to the working condition without accumulator.

In engineering, the pre-inflation volume of an accumulator is determined by its geometric parameters, and the only parameter that can be easily adjusted for existing accumulators is the pre-inflation pressure. For the single accumulator system determined in the previous part, a series of pre-inflation pressures were set for buffering performance estimate.

As shown in Fig. 5, when the external impact displacement is $0.25\,{\text{m}}$ and 0.5 m, the optimal pre-inflation pressures are $3.7\,{\text{MPa}}$ and $2.1\,{\text{MPa}}$, respectively. At this time, the buffering performance of the whole system is the best, reaching $39.7\%$ and $49.8\%$ respectively.

Fig. 5

Single accumulator buffering test

For accumulator group combination system, the first to consider is that as the number of accumulators increases, there could be different series and parallel combinations, which may affect the overall performance of the accumulator group. As shown in Fig. 6, for situations where the arrangement is on the same main pipelines, these three series and parallel combinations are essentially equivalent without significant influence on the overall performance of the accumulator group.

Fig. 6

The different combination of three accumulators

When designing the number of accumulators in an accumulator group under the same pre-inflation pressure, it is important to consider that their buffering efficiency does not differ significantly at each load pressure, in order to achieve a globally balanced and optimal buffering performance of the accumulator group. A combination of 1 to 6 accumulators was set up for this purpose, selecting a pre-inflation pressure of $9\,{\text{MPa}}$, with an external impact displacement of $0$ to $0.5\,{\text{m}}$, and each interval of $0.02\,{\text{m}}$ was used to load the system with one impact. The experimental results are shown in Fig. 7.

Fig. 7

Accumulator group buffering test

From Fig. 7, it can be seen that when the external impact displacement is below about 0.22–0.26 m, the fewer the number of accumulators, the better the overall system buffering performance. Moreover, at around $0.1\,{\text{m}}$, the global performance of a single accumulator reaches its optimal level, which is about $13.2\%$. When the external impact displacement is higher than $0.25\,{\text{m}}$, as the number of accumulators increases, the overall system buffering performance increases and tends to a stable value. At the maximum impact displacement, the final buffering performance of the six accumulators is about $9\%$, much higher than the $4.4\%$ of single accumulator.

4 Discussion and Conclusion

The simulation results in Figs. 5 and 7 can be summarized as follows: As the final design parameter for the accumulator group, each pre-inflation pressure corresponds to an external impact displacement range where the accumulator has the optimal buffering performance. In addition, under the same pre-inflation pressure, the number of accumulators and buffering performance are not directly proportional. When the external impact is small, an increase in the number of accumulators will actually reduce the overall buffering performance of the system.

Based on this issue, a concept of establishing a combination design program is proposed, which combines the combination optimization algorithm to design an optimization program for pre-inflation pressure and the number of accumulators, so as to achieve a global equilibrium and optimal buffering performance of the accumulator group. In actual algorithm selection, genetic algorithm, ant colony algorithm, or simulated annealing algorithm [8] can be used. The program diagram is shown in Fig. 8. Record the number of accumulators as $N$, and the pre-inflation pressure of each accumulator as ${P}_{i}$.

Fig. 8

Combination design program chart

Based on the actual parameters of a type of heavy-load transport trailer, the design of accumulator combinations was studied. A mathematical model of a single accumulator system was established, and based on this, an AMESim simulation model of a combined accumulator system was established.

With a series of tests conducted on the simulation model, the comprehensive effects of the number of accumulators in the group and the pre-inflation pressure on the performance of the accumulator group were found. In the discussion section, the comprehensive impact was evaluated and the process of combination design was obtained. The system performance could be increased by configuring suitable number of accumulators with each different parameter configurations.