Analysis of stiffness characteristics of a new fluid bag for axial shock protection

2.1. The fluid bag model

The fluid bag, which is a flexible sealed container, is full of fluid. It is a new kind of nonmetal spring working through the approximate incompressibility of fluid. The diagram of fluid bag is shown in Fig. 1, the outside part is bag cloth, and the inside part is water. The pressure of fluid bag can be adjusted to the predetermined value.

Fig. 1The diagram of fluid bag model

2.2. The application of fluid bag

The diagram of a mechanism which requires axial protection is shown in Fig. 2. The inner and outer shell are glued together through the rubber joints. The steel and rubber are vulcanized to the joints, which is similar to the rubber springs for rolling stock [20]. Large axial upward shock loads are imposed on the bottom of inner shell when the mechanism works. However, as for the inner shell, large axial static displacement is not allowed, otherwise the adhesive joints will fail to work. In order to avoid it, a fluid bag which is able to decrease the loads transmitted to the joints, was placed between the inner and outer shell, as is shown in Fig. 3.

Fig. 2The diagram of mechanism

Fig. 3The diagram of mechanism with a fluid bag

For the purpose of obtaining its axial protection performance, the axial stiffness characteristics of fluid bag will be analyzed alone. The simplified model for analyzing the axial stiffness is shown in Fig. 4. The outer shell was fixed, and the inner shell was adjusted to a predetermined location to be fixed in the axle. The fluid bag pressure was adjusted to a predetermined value. Then the inner shell was set up to be free and large axial loads were applied to its bottom region. The inner and outer shell models are shown in Figs. 5 and 6.

Fig. 4The diagram of simplified model

Fig. 5The diagram of inner shell model

Fig. 6The diagram of outer shell model

While the axial loads are transmitted from the inner shell to rubber joints, the fluid bag are compressed first. Due to the approximate incompressibility of fluid, large loads are generated by the bag under small axial displacement and deformation, avoiding the large static displacement in rubber joints.

3.1. The simulation of fluid bag

In Abaqus, there are a family of elements being able to represent fluid-filled cavities under hydrostatic conditions. The elements provide the coupling between the deformation of fluid-filled structure and the pressure exerted by fluid on the boundary of cavity [21]. When the fluid bag is compressed, the volume and pressure of fluid will change with its deformation. The volume function of fluid is given as:

where $p$ is the initial fluid pressure, $\theta$ is the fluid temperature, $m$ is the fluid mass. Due to the bag being full of fluid, the fluid volume is always equal to the bag volume. So:

where $V$ is the volume of bag cavity.

When the bag deforms, the volume and pressure of fluid will change. When the fluid pressure equals to p, the virtual work contribution due to the fluid pressure is given as:

where $\delta T^{*}$ is the increment of virtual work and $\delta T$ is the virtual work for the bag without the cavity. The negative signs mean that an increase in the bag volume releases energy from the fluid. This indicates a mixed formulation in which the bag displacement and fluid pressure are the two primary variables. The rate of the augmented virtual work expression is shown as:

Eq. (4) can be transformed into:

where $–pd\delta V$ indicates the pressure load stiffness, and $d\stackrel{-}{V}/dp$ represents the volume-pressure compliance of fluid.

Since the pressure is the same for all the elements of bag and fluid, the virtual work contribution can also be written as:

Eq. (6) can be transformed into:

where $T^a$ means the virtual strain energy of an element, $V^a$ means the volume a fluid element accounts for in the bag when it is compressed, and ${\stackrel{-}{V}}^a$ means the volume of a fluid element.

Since the temperature is the same for all the elements of fluid, the volume of every fluid element can be expressed as:

where $m^a$ means the mass of an element.

When the external load or fluid pressure changes, so will the shape of bag and fluid. Therefore, the volume a fluid element occupies in the bag may be different from the actual volume of a fluid element. Thus:

However, the total fluid volume is always equal to the bag volume.

To simulate the real mechanical characteristics of fluid bag is to simulate the coupling between the bag and fluid. There are hydrostatic elements being approximate incompressible in Abaqus. These elements can simulate the coupling between the bag deformation and fluid pressure. The fluid mass $m$ is related to its density$\mathrm{\rho }$, so the variable $m$ can be indicated by $\rho$. Considering the density, temperature and pressure is respectively uniform, a reference node has been set for all the fluid elements, and the coupling between the reference node and every element has been built. Then the density, temperature and pressure of fluid have been set in the reference node. Thus the real simulation of fluid bag has been completed.

3.2. The finite element model

Due to the axial and circumferential size of mechanism being much bigger than its thickness, shell elements were used for the simulation. The S4R finite-strain shell elements were used for the bag, the inner and outer shell. In an attempt to simulate the coupling between the bag and fluid, the fluid elements shared the same nodes with the bag elements. To be specific, the bag elements were duplicated. Then the duplicated elements were changed into F3D4 hydrostatic elements, the element labels being changed and node labels remaining unchanged. The finite element model is shown in Fig. 7.

Fig. 7The finite element model of mechanism

3.3. The simulation of contact

While being compressed, the bag will contact with inner and outer shell. The contact states changes along with the variation of bag shape. However, the rule of this variation is beyond prediction. Therefore, it is necessary to establish real contact pairs. Surface with coarse meshes, a large area and high stiffness are always the master surfaces, and soft surface with a small area and fine meshes are always the slave surfaces [22]. There are two contact pairs in this paper. The first contact pair, the outer surface of inner shell being the master surface and inner diameter surface of bag being the slave surface, is shown in Fig. 8. The second contact pair, the inner surface of outer shell being the master surface and outer diameter surface of bag being the slave surface, is shown in Fig. 9. To ensure an easy convergence for the calculations, tolerance of 0.1 mm has been specified for the two contact pairs.

Fig. 8The master surface and slave surface for first contact pair

Fig. 9The master surface and slave surface for second contact pair

3.4. The boundary conditions and material properties

According to chapter 2.2, the outer shell is fixed. For the purpose of axial loads being imposed easily, the continuous distributing coupling was established between the coordinate origin and the bottom surface of inner shell, making the loads imposed on the coordinate origin distribute on the bottom surface of inner shell. The boundary conditions and load coupling are shown in Fig. 10.

Fig. 10The diagram of boundary conditions and load coupling

The material of inner and outer shell is high-strength steel. The bag material is rubber fiber, and the bag is full of water. The equivalent elastic modulus, Poisson’s ratio and density are given in Table 1.

3.5. The experimental verification of axial stiffness characteristics

Due to the fluid bag providing axial protection for rubber joints, the whole stiffness of fluid bag and joints were measured in experiments. Then the stiffness of fluid bag could be derived from the stiffness of joints measured in advance. The inner shell would be restricted by the joints while moving down in the axle. Therefore, its position does not need to be adjusted while the bag are being filled with fluid.

3.5.1. The introduction of experiment

The schematic is shown in Fig. 11, where the inner and outer shell are connected by the rubber joints as displayed. The fluid bag lies between the inner and outer shell. Four brackets, being made of steel, are bolted with the outer shell and pinned to the ground uniformly. A piece of load transfer plate is attached and bolted to the inner shell, and is connected to the hydraulic cylinder.

Fig. 11The experimental schematic diagram

A pressure sensor is established at the orifice of the fluid bag for the purpose of measuring the pressure inside the fluid bag. Four displacement sensors are placed on the edge of the bottom of the inner shell, which will offer an average displacement value for the experiment. The pressure sensor, displacement sensor, dynamic testing instrument and hydraulic cylinder is respectively shown in Fig. 12 to Fig. 15.

The form of load in the experiment is shown in Fig. 16.

Fig. 12The pressure sensor

Fig. 13The displacement sensor

Fig. 14The dynamic testing instrument

Fig. 15The hydraulic cylinder

Fig. 16The curve of load varying with time

The experiment procedures are listed below.

Step 1, the stiffness of the rubber joints without the fluid bag was measured. The load was applied as required, and the displacement of the edge of the bottom of the inner shell along the axial direction was measured.

Step 2, the fluid bag was placed between the inner and outer shell, and was pressurized to 1.5 MPa. The air in the bag being exhausted as far as possible, the load was applied according to the requirement, and the displacement of the edge of the bottom of the inner shell along the axis direction as well as the pressure in the fluid bag was tested.

Step 3, the initial pressure of the bag was set to 2.0 MPa, then other procedures in Step 2 were repeated.

Step 4, according to the data from last three steps, the displacement-load curve and pressure-load curve were formed. Afterwards, the least square method was used to form the linear fitting chart and the slopes of the lines were acquired.

Step 5, from Step 4 the axial stiffness of the rubber joints and the axial stiffness of the fluid bag together with the rubber joint were acquired.

3.5.2. The comparison between simulation and experiment results

When the initial bag pressure is 1.5 MPa, the simulation and experiment results are shown in Fig. 17 and Fig. 18.

When the initial bag pressure is 2.0 MPa, the simulation and experiment results are shown in Fig. 19 and Fig. 20.

Fig. 17The diagram of whole axial stiffness of simulations and experiments

Fig. 18The diagram of bag pressure of simulations and experiments

Fig. 19The diagram of whole axial stiffness of simulations and experiments

Fig. 20The diagram of bag pressure of simulations and experiments

According to Fig. 17 to Fig. 20, the curves of axial stiffness and pressure variations have good linearity. The difference of results between the simulations and experiments is small, proving the correctness of simulations.

Table 2The whole axial stiffness under different initial bag pressures

The bag pressure / MPa	The axial stiffness of inner shell (kN/mm)
	The simulation result	The experimental result
1.5	224.3	234.7
2.0	222.8	231.6

As for the rubber joints, the elastic modulus of steel is 210000 MPa, and the equivalent modulus of rubber is 1.25 MPa. The measured axial stiffness of rubber joints is 74.2 kN/mm. Therefore, the measured axial stiffness of fluid bag is 160.5 kN/mm under 1.5 MPa initial pressure, and the value would be 157.4 kN/mm under 2.0 MPa initial pressure.

According to Fig. 18 and Fig. 20, the fluid bag is compressed by the inner shell with the increasing axial load. Due to the bag being full of approximately incompressible fluid, the bag pressure increases rapidly during the progress. Most of the loads on the inner shell could be transmitted to outer shell through the fluid bag, providing axial protection for the rubber joints. And the bag pressure variations of simulations and experiments are very similar, which demonstrates that the simulations hold water.

4.1. The axial stiffness of fluid bag under different initial pressures

The results are displayed in Fig. 23 to Fig. 27 when the initial pressure is respectively 1.5 MPa, 2.0 MPa, 2.5 MPa, 3.0 MPa and 3.5 MPa.

Fig. 23The diagram of axial displacement and load of inner shell

Fig. 24The diagram of bag pressure and load

According to Fig. 23, the axial stiffness of fluid bag remains basically unchanged with load increasing when an initial pressure is exerted. The axial stiffness remains basically unchanged when the initial pressure turned from 1.5 MPa to 3.5 MPa. According to the calculations, while the initial pressure varies from 1.5 to 3.5 MPa, the corresponding stiffness are 160.93, 158.95, 156.91, 154.98 and 153.12 kN/mm, which are extremely close to the experimental results.

According to Fig. 24, the initial loads are different, owing to the fluid bag having different balanced loads when different initial pressures are exerted. The bag pressure increases along with load, and the pressure variation tendency has tiny nonlinearity. For example, when the load is 177.28 kN and initial pressure is respectively 1.5 to 3.5 MPa, the corresponding bag pressure is 3.23, 3.30, 3.38, 3.44 and 3.50 MPa, and the pressure difference of adjacent curves are respectively 0.079, 0.072, 0.067 and 0.058 MPa. When the load increases to 497.28 kN, the corresponding bag pressures are 7.91, 8.09, 8.27, 8.42 and 8.56 MPa, and the pressure difference of adjacent curves are respectively 0.179, 0.172, 0.156 and 0.141 MPa. The pressure difference of adjacent curves increases with load when different initial pressures are exerted.

Fig. 25The diagram of inner contact area and load

Fig. 26The diagram of outer contact area and load

In Fig. 25 and Fig. 26, the inner contact area means the contact area between inner shell and fluid bag, and the outer contact area means the area between outer shell and fluid bag. According to the figures, the inner and outer contact area both increase along with load, after reaching their respective maximum values, they are likely to almost remain unchanged under different initial pressures, except for 1.5 and 2.0 MPa. Owing to being relatively soft, the bag is gradually compressed by the increasing load, making the contact area gradually expand. Due to being not able to be compressed unlimitedly, the contact area will remain basically unchanged after reaching their peak figures. While the initial pressure is respectively 1.5 and 2.0 MPa, the contact area shrinks with load after reaching their peak figures. According to Fig. 22, due to the fluid bag moving downward with load, the low part of bag would no longer contact with the inner and outer shell after the contact area reaching peak figures, which eventually leads to the reduction of the area of contact.

The corresponding contact area decreases with the initial pressure. The higher initial pressure, as well as the large amount of fluid contained by the bag, is likely to reduce the initial contact area. However, when the contact area reaches peak figures, the corresponding load would increase with initial pressure. It can be inferred that the higher the initial pressure is, the stronger the bag’s ability to resist deformation is.

Fig. 27The diagram of bag volume and load

According to Fig. 27, the bag volume decreases with load, and the variation tendency remains basically unchanged under different initial pressures.

4.2. The axial stiffness of fluid bag under different material properties

The initial pressure of fluid bag is 2.5 MPa, and the fluid bulk modulus is 2200 MPa. The results are shown in Fig. 28 and Fig. 29 when the elastic modulus of bag is respectively 4000, 5000, 6000, 7000, 8000 and 9000 MPa.

Fig. 28The diagram of axial displacement and load of inner shell

Fig. 29The diagram of bag pressure and load

According to Fig. 28, the axial stiffness of fluid bag is evidently different under different elastic modulus. The axial stiffness increases with elastic modulus, but the trend of rising grows weaker afterwards. For example, when the elastic modulus of bag varies from 4000 to 9000 MPa, the corresponding stiffness are 101.84, 121.74, 140.03, 156.91, 172.55 and 187.09 kN/mm, and the stiffness differences between the adjacent curves are respectively 19.9, 18.29, 16.88, 15.64 and 14.54 kN/mm.

According to Fig. 29, all the curves almost coincide with one another, indicating that the difference of elastic modulus basically has no effects on the bag pressure variation.

4.3. The axial stiffness of fluid bag under different fluid bulk modulus

The initial pressure of fluid bag is 2.5 MPa, and the elastic modulus of bag is 7000 MPa. The results are shown in Fig. 30 and Fig. 31 when the fluid bulk modulus is respectively 200, 400, 600, 1000, 1400, 1600, 1800, 2000, 2200 and 2400 MPa.

Fig. 30The diagram of axial displacement and load of inner shell

Fig. 31The diagram of the pressure and load

According to Fig. 30 and Fig. 31, the axial stiffness and bag pressure increase with load, but the increasing trends are becoming weaker.

The axial stiffness and pressure variation of fluid bag have bad linearity when the fluid bulk modulus is small. It is indicated that the fluid is easy to be compressed under small bulk modulus, making the bag have relatively large deformation. However, with the fluid bulk modulus increasing gradually, the fluid becomes difficult to be compressed, making the deformation progress harder to occur on the bag. Thus the curves have relatively good linearity.

4.4. The finite element analysis of fluid bag for axial protection

According to Chapter 2.2, the inner and outer shell are glued together through the rubber joints. And the inner shell will have a big impact upward in the axle when the mechanism works. If the stress in the rubber joints is too large, it may lead to the mechanism failing to work. Therefore, it is necessary to decrease the stress in the joints.

Fig. 32The finite element model of mechanism

Fig. 33The finite element model of mechanism with a fluid bag

In order to obtain the bag’s ability to decrease the stress, a simplified shock model has been established. Considering the exactitude and efficiency in calculation, the rubber joints are simplified as a rubber pad. Due to the focus of research being the bag’s ability to decrease the stress, the modeling of adherend layer is ignored. The exact model is shown in Fig. 32. It is assumed that the outer shell is stationary while the load is transmitted upward. Thus the outer shell is fixed. The finite element model of mechanism with a fluid bag is shown in Fig. 33. The joints, inner and outer shell are joined together with tie constraints. The material properties are shown in Table 1, and the boundary conditions are shown in Fig. 10.

When the load is 250 kN, the stress distribution between the joints and inner shell is shown in Fig. 34 to Fig. 37.

According to Fig. 34 to Fig. 37, the stress distribution trends are basically the same whether mechanism has a bag or not. When it comes down to the mechanism without a bag, the maximum stress between the bag and inner shell is 5.525 MPa, and the value between the bag and outer shell is 6.066 MPa. While the mechanism has a bag, the value between the bag and inner shell becomes 2.449 MPa, and the value between the bag and outer shell becomes 2.689 MPa. It is indicated that the bag can decrease the stress in the joints effectively, having a good performance for axial protection.

Fig. 34The diagram of stress between the joints and inner shell without the fluid bag

Fig. 35The diagram of stress between the joints and outer shell without the fluid bag

Fig. 36The diagram of stress between the joints and inner shell with a fluid bag

Fig. 37The diagram of stress between the joints and outer shell with a fluid bag