Keywords

1 Introduction

Hydraulic shock absorber are an important component of automotive suspension systems [1,2,3], which can convert mechanical energy into thermal energy, thereby attenuating the vibration and shock caused by road excitation and improving the handling stability of the vehicle. However, the hydraulic shock absorber is subjected to frequent, high-speed vibration impacts during which the shock absorber fluid dissolves into air bubbles and cavitation occurs [4,5,6,7]. This phenomenon reacts to the shock absorber operation called shock absorber cavitation, which can lead to the failure of the shock absorber in a certain section of the stroke, causing an increase in vehicle vibration, or a bird-like noise, and the cavitation phenomenon can reduce the life cycle of the shock absorber [8]. Therefore, it is significant to study the air travel phenomenon of hydraulic shock absorbers.

At present, there are few studies on the cavitation phenomenon of the shock absorber, especially the detailed theory and the location where it occurs. In this paper, to solve this problem of the cavitation phenomenon of the shock absorber, high-precision hydraulic shock absorber solid and fluid finite element models are established respectively, and the finite element models are simulated in ADINA software to determine the specific location of the cavitation phenomenon inside the shock absorber. At the same time, the influence of different fluid parameters on the hydraulic shock absorber cavitation phenomenon was analyzed, and effective measures and methods to weaken the cavitation phenomenon were proposed and verified through experiments. The cavitation phenomenon of the hydraulic shock absorber is weakened.

2 Analysis of the Causes of Hydraulic Shock Absorber Loss Stroke

2.1 Shock Absorber Oil Cavitation Principle

Since the shock absorber is equipped with shock absorber oil inside, there is a certain amount of gas mixed in the oil, and as the pressure and temperature field inside the shock absorber changes, the air dissolved inside the oil will increase or decrease [9, 10].

As shown in Fig. 1, the shock absorber produces a cavitation phenomenon: the shock absorber recovery valve will be divided into the upper and lower chambers of the shock absorber, when the piston rod upward or downward movement, the upper and lower cavity fluid is compressed, the oil pressure becomes larger, this cavity fluid dissolved into more gas, when the throttle piece open valve pressure, the pressure of the compressed chamber instantly decreases, when the pressure drops to the saturated vapor pressure of the fluid, the fluid has been pressed into the gas will be released, so that the shock absorber fluid produces a large number of bubbles, this phenomenon for the hydraulic shock absorber cavitation phenomenon [11].

Fig. 1
A schematic of a double-cylinder oil-hydraulic vehicle shock absorber with the internal structure and functioning components including the upper and lower chambers, piston rod, tube reservoir, pressure tube, and valves for rebound and compression.

Schematic diagram of the double cylinder oil-hydraulic vehicle shock absorber 1. Piston rod; 2. Tube reservoir; 3. Pressure tube; 4. Valves SA-rebound; 5. Valves SA-compression

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2.2 Parametric Model of Shock Absorber Cavitation [12]

At constant temperature, the cavitation coefficient \(\sigma\) is usually used to characterize the extent to which cavitation occurs. It is expressed by Eq. (1).

$$\sigma = \frac{{P_{2} - P_{{\text{v}}} }}{{{{\rho v^{2} } \mathord{\left/ {\vphantom {{\rho v^{2} } 2}} \right. \kern-0pt} 2}}}$$
(1)

P2—shock absorber lower cavity pressure; Pv—shock absorber fluid cavitation critical pressure; ρ—shock absorber fluid density; v—fluid flow average speed.

When the hydraulic shock absorber moves up and down, the shock absorber fluid passes through the recovery valve system and reciprocates between the upper and lower chambers of the shock absorber, generating a differential pressure ΔP.ΔP can be defined as:

$$\Delta P = P_{1} - P_{{2}} = {{\rho v^{2} } \mathord{\left/ {\vphantom {{\rho v^{2} } 2}} \right. \kern-0pt} 2}$$
(2)

P1-shock absorber upper chamber pressure. Pv-the critical pressure of shock absorber fluid cavitation，can be neglected compared to P2 and P1, i.e., Pv = 0, which can be simplified by Eqs. (1) and (2) as:

$$\sigma = \frac{{P_{2} }}{{P_{1} - P_{2} }}$$
(3)

When \(\sigma\) is less than 0.4, cavitation phenomenon occurs [13]. \(\sigma\) = 0.4 is the critical point at which the cavitation phenomenon occurs in the oil-hydraulic shock absorber. As a result, Eq. (3) is simplified as.

$$\xi = \frac{{P_{1} }}{{P_{2} }} = 3.5$$
(4)

ξis the oil-hydraulic shock absorber cavitation critical pressure ratio.

3 FSI Finite Element Analysis Method Based on ADINA

In order to more intuitively observe the cavitation phenomenon of the oil-hydraulic shock absorber, this paper establishes a finite element model of the cavitation phenomenon generated by the oil-hydraulic shock absorber. When the host plant tests the shock absorber, the common test temperature requirement is 22 degrees Celsius, so this paper selects this temperature value for the saturation vapor pressure of the shock absorber fluid.

3.1 FSI Finite Element Mathematical Model [14]

FSI finite element analysis is used to solve the problem of the state of motion between a fluid and a solid, and its basic principle is to satisfy both the fluid-coupled boundary dynamics and kinematic equations, and the solid-coupled boundary dynamics and kinematic equations.

$$d_{f} = d_{s}$$
(5)

The kinetic conditions:

$$n \cdot \tau_{f} = n \cdot \tau_{s}$$
(6)

where df and ds are the fluid and solid boundary displacements, respectively, τf and τs are the fluid and solid stresses, respectively, and n is the unit external normal vector. From the kinematic conditions, the fluid velocity conditions can be derived as follows.

$$n \cdot v_{f} = n \cdot {\text{v}}_{s}$$
(7)

3.2 ADINA-based Flow-solid Coupling FEM Analysis

When using the finite element FSI theory to analyze the flow-solid coupling problem of the oil-hydraulic shock absorber, two key points need to be addressed.

  1. (1)

    Accurate transfer of interaction forces on the fluid–solid coupling surface.The solid model and fluid model of ADINA software are established separately, and the boundary conditions are defined on the coupling surfaces of the two models to solve the flow-solid coupling problem. The meshes of the fluid and solid do not need to be identical, as long as a certain tolerance is achieved [15]. Therefore, the fluid nodal displacements on the fluid–solid coupling surface are obtained by interpolation of the nearby solid nodal displacements, while the solid nodal forces are obtained by integration of the surrounding fluid stresses, i.e.

    $$F(t) = \int {h^{d} \tau_{f} dS}$$
    (8)

    Where hd is the solid imaginary displacement

  1. (2)

    Realizing efficient solution of coupled systems ALE (Arbitrary Lagrangian Euleria) is currently used to solve the problem of fluid mesh movement due to solid deformation.

ADINA software has both iterative and direct solution methods, both of which need to ensure the consistency of the integration points of the fluid and solid models during dynamic analysis. Both methods require iterative solutions in order to solve the fluid and solid equations separately. The direct solution method puts the fluid model and the solid model in the same matrix, and the finite element equations are as follows:

$$\left[ \begin{gathered} A_{ff} \begin{array}{*{20}c} {} & {} \\ {} & {} \\ \end{array} A_{fs} \hfill \\ A_{sf} \begin{array}{*{20}c} {} & {} \\ {} & {} \\ \end{array} A_{ss} \hfill \\ \end{gathered} \right]\begin{array}{*{20}c} {} \\ {} \\ \end{array} \left[ \begin{gathered} \Delta X_{f}^{k} \hfill \\ \hfill \\ \Delta X_{s}^{k} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} B_{f} \hfill \\ \hfill \\ B_{s} \hfill \\ \end{gathered} \right]$$
(9)
$$X^{k + 1} = X^{k} + \Delta X^{k}$$
(10)
$$\begin{array}{*{20}c} {A_{ff} = \frac{{\partial F_{f}^{k} }}{{\partial X_{f} }}} & {A_{fs} = \lambda_{d} \frac{{\partial F_{f}^{k} }}{{\partial X_{s} }}} \\ \end{array}$$
(11)
$$\begin{array}{*{20}c} {A_{sf} = \lambda_{\tau } \frac{{\partial F_{f}^{k} }}{{\partial X_{f} }}} & {A_{ss} = \frac{{\partial F_{s}^{k} }}{{\partial X_{s} }}} \\ \end{array}$$
(12)

where the displacement relaxation factor is λd（0＜λ d ≤ 1）and the stress relaxation factor is λτ(0＜λτ ≤ 1） the direct solution method has high accuracy when solving non-contact transient models where the computational volume is not very large.

3.3 Establishment of the Finite Element Model of the Oil-hydraulic Shock Absorber Superimposed Throttling Recovery Valve System

In order to build a superimposed throttling recovery valve system for the oil shock absorber, the overall valve system needs to be simplified in consideration of the accuracy of the finite element model and the computing capability of the computer.

  1. (1)

    Since the valve system is a symmetric structure, a 1/4 finite element model can be built to meet the requirements.

  2. (2)

    The sum of the multi-groove valve sheet area is converted into contact area, which does not affect the calculation accuracy.

  3. (3)

    The piston chamfer of the valve system is removed.

  4. (4)

    The purpose of the simulation is to observe the cavitation phenomenon inside the oil shock absorber, so the valve piece and nut are removed.

Figure 2 shows the fluid finite element mesh model, and Fig. 3 shows the solid finite element mesh model. The fluid model is an 8-node hexahedral cell with all variables defined at each corner node. This cell satisfies the inf–sup stability and optimality conditions, and has good second-order accuracy when calculating variable interpolation, which is more accurate than the first-order accuracy of the tetrahedral cell. Considering the computational power of the computer and the solution time, the mesh is subdivided only in the multi-piece throttle piece and the fluid coupling part, which can ensure that the relative displacement is within the tolerance range when the fluid–solid coupling surface is calculated. To ensure the density compatibility of the fluid–solid coupling surface, the solid model is subdivided by 8-node hexahedral unitization. The solid model is divided into 8-node hexahedron cells, with 11,277 solid meshes and 29,733 fluid meshes, to ensure density compatibility of the fluid–solid coupling surface.

Fig. 2
An illustration has two uneven structures connected via a cylindrical rod. Each structure has mesh and grid patterns.

Fluid finite element model

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Fig. 3
An illustration of a solid finite element model has stacked uneven structures with mesh and grid patterns.

Solid finite element model

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The velocity is applied at the fluid inlet of the recovery valve, the loaded velocity is equal in magnitude to the piston moving velocity but in the opposite direction, and the loaded amplitude is 1/4 sine wave of 1 m/s for 1.56s, as shown in Fig. 4. No boundary conditions are set at the outlet of the fluid model, and the default pressure value is zero. Set the fluid–solid coupling surface boundary conditions and set to symmetric properties, the rest of the fluid boundary are set to wall, so set because the sliding interface of the FCBI unit, can only be solved with a sparse solver, and multiple throttle pieces in the fluid motion will produce deformation, is a transient nonlinear motion, in the ADINA software using the leader–follower command, this command can make the fluid flow The leader–follower command is used in the ADINA software to keep the fluid flow in the same direction, and the integration can be calculated for the entire fluid level in post-processing.

In the solids model, there are five valves forming a group of stacked multi-piece throttling valves, which are defined as contact between the valves, and the entire boundary of the stacked multi-piece throttling valves is defined as contact with the piston. The implicit dynamic method is set as the solids analysis type.

3.4 Analysis of Simulation Results

The simulation results are shown in (Figs. 4 and 5).

Fig. 4
A graph of time function value versus time plots a concave downward ascending curve.

Fluid velocity loaded sine wave

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Fig. 5
A cloud diagram of the oil shock absorber plots the distribution of effective stress using a color gradient scale. Majority of the distribution is uniform at 10000. A portion at the bottom has stress varying from 50000 to 90000.

Oil shock absorber pressure cavitation cloud diagram t = 0.66 s, v = 0.42 m/s

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Figure 5 for t = 0.66 s, v = 0.42 m/s, v = 0.77 m/s, and 1 m/s. The values are all sinusoidal, with an amplitude of 30 mm, corresponding to t = 0.66 s, t = 1.2 s, and t = 1.56 s. The simulation results are shown in Fig. 5 for t = 0.66 s, v = 0.42 m/s, and the graph shows that at this time the valve has just been opened, and the pressure drop is not obvious. So the cavitation phenomenon is not obvious, at this time the oil shock absorber work in the normal state. As shown in Fig. 6, t = 1.2 s, v = 0.77 m/s, as can be seen in the figure, at this time the valve has opened, the pressure drop is larger, at this time the cavitation phenomenon around the valve system is clearly presented, at this time the oil shock absorber work destabilization is not obvious. As shown in Fig. 7, t = 1.56 s, v = 1 m/s, at this time the superimposed throttle valve system completely open, the pressure drop is the largest, the oil shock absorber cavitation phenomenon is the most obvious, cavitation phenomenon with the valve system mouth down the cavity to move, at this time the oil shock absorber work part of the stroke failure state. From Figs. 5, 6 and 7 can be seen, the cavitation phenomenon initially.

Fig. 6
A cloud diagram of the oil shock absorber plots the distribution of effective stress using a color gradient scale. Majority of the distribution is uniform at 16667. A portion at the bottom has stress varying from 50000 to 216667.

Oil shock absorber pressure cavitation cloud diagram t = 1.2 s, v = 0.77 m/s

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Fig. 7
A cloud diagram of the oil shock absorber plots the distribution of effective stress using a color gradient scale. Majority of the distribution is uniform at 18000. A portion at the bottom has stress varying from 54000 to 234000.

Oil shock absorber pressure cavitation cloud diagram t = 1.56 s, v = 1 m/s

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distributed around the superimposed throttle piece, the formation of the lower cavity spread, with the oil-hydraulic shock absorber piston speed increases, the more obvious the shock absorber cavitation phenomenon.

4 Fluid Viscosity on the Shock Absorber Air Range Phenomenon of the Impact

The lower the kinematic viscosity of the shock absorber fluid, the smaller the pressure loss, the less likely it is to be below the saturated vapor pressure of the fluid, and the less obvious the oil shock absorber air travel phenomenon.

In this paper, three different kinematic viscosities of the shock absorber fluid are used for simulation comparison, the piston hole diameter is 1.5mm, and the fluid viscosities are 12.3 mm2/s, 13.3 mm2/s and 14.3 mm2/s respectively. The simulation results are compared under the same conditions.

The simulation results are shown in Fig. 8. When the kinematic viscosity is 14.3 mm2/s, the cavitation phenomenon is more pronounced than the other two kinematic viscosities in the three time regions of t = 0.66 s, t = 1.2 s, and t = 1.56 s, proving that the high kinematic viscosity of the shock absorber fluid is prone to produce the shock absorber cavitation phenomenon. When the oil kinematic viscosity is 12.3 mm2/s, in the three time regions of t = 0.66 s, t = 1.2 s and t = 1.56 s, respectively, compared with the other two oil kinematic viscosities, the shock absorber cavitation phenomenon is the least obvious, indicating that the lower the oil viscosity, the less obvious the shock absorber cavitation phenomenon, indicating that low viscosity oil has a certain effect on reducing the shock absorber cavitation phenomenon.

Fig. 8
3 sets of 3 cloud diagrams of the oil shock absorber plot the distribution of effective stress using a color gradient scale. Majority of the distribution is uniform at lower values. A portion at the bottom has stress varying in higher value ranges.

Pressure cavitation cloud of shock absorber oil with different kinematic viscosities. a Oil kinematic viscosity of 14.3 mm2/s. b Oil kinematic viscosity of 13.3 mm2/s. c Oil kinematic viscosity is 12.3 mm2/s

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5 Conclusion

  1. (1)

    Based on the ADINA numerical method, the cavitation phenomenon of the hydraulic shock absorber throttle superimposed valve system was simulated, and the simulation clouds observed that the specific location where cavitation was generated was mainly in the vicinity of the shock absorber throttle valve system.

  2. (2)

    The finite element mesh of the shock absorber throttle system is established, and the solid and fluid hexahedral meshes are established respectively. The mesh is calculated with second-order accuracy, and the model is close to the actual physical model, and the Euler–Lagrange model is used to solve the problem. The low-speed multi-groove valve piece is replaced by a constant through-hole, and both flow rates are kept the same. The simulation model is as close as possible to the actual, so that the calculation results are accurate.

  3. (3)

    It is proposed that the shock absorber fluid kinematic viscosity has a certain effect on attenuating the shock absorber cavitation phenomenon. The simulation results indicating that the cavitation phenomenon becomes more obvious with the increase of piston speed and the lower the kinematic viscosity of the shock absorber oil, the less obvious the cavitation phenomenon.