Vibration isolation characteristics of a rubber isolator in the deep water condition

In order to explore the effect of the deep water condition on the vibration isolation performance of rubber isolators, a finite element model of shipboard rubber vibration isolator with different depths which considers the nonlinear material properties of rubber is established. The nonlinear properties of rubber materials are characterized by the Mooney-Rivlin hyper-elasticity model. Hydrostatic pressure is applied on the surface of the isolator in the form of pre-stress to simulate deep water conditions. The static and dynamic mechanical characteristics of the rubber isolator under deep water environment are obtained. The results show that the static stiffness of vibration isolator increases with the water depth, especially when the rubber hardness is high. The natural frequency of the isolator under deep water environment increases with the water depth. The transmission loss of rubber isolators under deep water environment significantly reduces due to the change of static stiffness. The results can provide technical support for the design and optimization of underwater vibration isolators.


Introduction
Underwater vehicles are subject to vibration loads generated by mechanical equipment such as motors carried on board during navigation.During underwater conditions, some underwater electronic equipment needs to be partially or even completely submerged in water.In order to control the vibration and noise of the vehicle and reduce the disturbance to the equipment, rubber isolators are often used for vibration reduction of the equipment.The static mechanical characteristics of rubber isolators play an important role in the smoothness and stable maneuverability of underwater equipment [1].Due to the different conditions of underwater equipment compared to land, rubber vibration isolators are directly exposed to the water.In deep water conditions, the hydrostatic pressure will have a significant impact on the vibration isolation characteristics of the rubber isolators, such as the static stiffness.Thus, the vibration isolation effect will be changed.
The rubber vibration isolator is widely used in various engineering fields.It is characterized by its compact structure, high damping and good workmanship.As a typical hyper-elastic material, rubber changes its shape while its volume remains unchanged during the force, accompanied by geometrical and material non-linearity.Nonlinear hyper-elastic mechanics is one of its most important basic properties.J.M. Hill summarized the analytical methods for solving the static characteristics of rubber isolators.The most representative of these methods is the static property calculation method based on the nonlinear shape factor method [2].M. Sass built biaxial experimental equipment and drew conclusions.It was concluded that the material model parameters obtained by fitting the data acquired from the two experiments can more accurately represent the mechanical properties of rubber hyper elasticity [3].Mille used uniaxial tensile, biaxial tensile and planar shear experimental equipment to improve the test specification and experimental data processing methods [4].Zhang et al. found that.When using the rubber hyper-elastic model to calculate the static stiffness of vibration isolators, it should minimize the error in fitting the experimental data to this model.
To solve this problem, a finite element model of shipboard rubber vibration isolators with different depths which selected four rubber materials with different Rockwell hardness and used the Mooney-Rivlin hyper-elastic model was established based on the structure of shipboard rubber vibration isolators.The vibration isolation characteristics of the rubber vibration isolators were simulated and analyzed under different deep water conditions.

Hyperplastic constitutive model for rubber materials.
Rubber materials present significant non-linearity in their stress-strain relationships throughout the deformation process [5].In order to describe the non-linear properties of rubber materials, it is necessary to develop hyper-elastic mechanical constitutive models for rubber materials.There are mainly two types of existing models [6].One is the statistical model based on molecular thermodynamics, including the Arruda-Boyce model and the Van der Waals model.The other type is the continuous medium mechanics models based on the image-only theory, including the Ogden model, the Mooney-Rivlin model, the Neo-Hooke model, the Yeoh model, the polynomial model and the reduced polynomial model [7].Central to the hyperplastic mechanical constitutive models for rubber materials is the expression of the strain energy function.The generic form of the strain energy function W is [8]: In the above equation,  1 ,  2 and J are the first, second and third order Green strain invariants respectively.They are all functions of the principal stretch ratios  1 ,  2 and  3 .
Here are the strain energy functions corresponding to several models.For rubber materials, the Mooney-Rivlin model strain energy function is: ( 2 − 3)  (2) In the above equation,   is the Rivlin coefficient,  00 = 0, i + j ≤ N , and N is the polynomial order.When N=1, equation (2) simplifies to the Mooney-Rivlin 2-parameter model strain energy function.
For rubber materials, the Yeoh model strain energy function is: (3) In the above equation,  10 ,  20 and  30 are parameters of the model related to temperature.
For rubber materials, the strain energy function for the 1st order Ogden model is: In the above equation, μ 1 and  1 are both related to temperature.For rubber materials, the strain energy function for the Arruda-Boyce model is: In the above equation,   is the thermodynamic parameter of the model and is calculated using statistical theory.G is the initial shear modulus of the rubber material.The coefficient   represents the lock-up strain [9].
FEM software is used to simulate and analyze the vibration isolation characteristics of rubber vibration isolators.In order to raise the accuracy of the calculation, a suitable hyper-elastic constitutive model should be selected.In this paper, the Mooney-Rivlin 2 model is used to analyze the rubber material.

Finite element modelling
A type of rubber vibration isolator widely used in ships is selected for vibration isolation characteristics study in this paper.As shown in figure 1.The vibration isolator consists of a metal support bearing, a metal plate and a rubber body.For the metal support bearing, the bottom part is covered by the rubber body and the top part is connected to the equipment being isolated.The plate is the metal structure of the vibration isolator.The rubber body is the main factor to determine the vibration isolation characteristics.The vibration isolator is simplified because of its structural complexity.The rubber vibration isolator model is cut into 1/4 models along the symmetry plane.The reason for this is that the geometric model, the loads and the boundary conditions are symmetrical.The rubber material is modeled using the constitutive model chosen above.The linear elastic material model is used for the metal material.The rubber and metal cells of the isolator are meshed using solid187.The mesh is shown in figure 2. There are in total 30,846 cells.Restraint and static loads are applied to the model.Different hydrostatic pressures are achieved in the form of prestress applied to the surface of the vibration isolator.

Static stiffness analysis
To study the effect of deep water conditions on rubber vibration isolators, four hardness of rubber are selected as rubber body materials.The four types of rubber are Rockwell hardness 75, 70, 65 and 60 respectively.Four water depth conditions are selected: 300m, 500m, 800m and 1000m.Simulation analysis is carried out for different hydrostatic pressure conditions.The static stiffness is calculated and compared with the static stiffness under atmospheric air pressure.The results are shown in figure 3    The relevant conclusions can be drawn from the above graphs.The stiffness of the vibration isolator increases with the increase in rubber hardness.The static stiffness of the vibration isolator increases significantly with an increase in hydrostatic pressure from 0Mpa to 10Mpa.This shows that the hydrostatic pressure has a significant effect on the static stiffness of the rubber vibration isolator.The change in stiffness varies with various rubber materials.This indicates that the effect of hydrostatic pressure on the static stiffness of the rubber isolator is influence affected by the rubber material.

Modal analysis
Following the above analysis of the static stiffness of the vibration isolators, the modal analysis is compared in this subsection.The results are shown in figure 5.As the hydrostatic pressure increases from 0Mpa to 10Mpa, the first order natural frequencies of the isolators with different rubber materials all increase by roughly 40%.This indicates that the natural frequency of the vibration isolator under deep water conditions increases with the water depth.This is a significant increase compared to the natural frequency of the isolator under atmospheric air.This finding is consistent with the stiffness changes obtained above.This shows that deep water conditions have a significant effect on the static dynamic characteristics of the isolator.

Vibration isolation characteristics analysis
In this subsection, the transmission loss of the vibration isolators will be analysed and compared at various water depths.The results are shown in figure 6.As seen in the above diagram, the vibration isolation effect of the vibration isolator decreases under deep water conditions.The transmission loss of the vibration isolator reduces with hydrostatic pressure.With the increase in hydrostatic pressure from 0 to 10 Mpa, the transmission loss reduces by around 4 to 7 dB.The vibration isolation effect of the isolators is affected by the different materials.As the rubber material chosen is softer, the less effect it will have.

Strength analysis
In addition to vibration isolation properties, the strength of the vibration isolator is analysed under different hydrostatic pressure conditions.The results are shown below.8 illustrates the maximum equivalent stresses of vibration isolators with different rubber materials under different water depth conditions.It increases with increasing water depth.As shown in figure 7, the maximum equivalent stresses generally appear at the bottom of the vibration isolator and where it is connected to the vibration-isolated object.

Conclusions
Based on a type of naval rubber vibration isolator, a 1/4 rubber vibration isolator finite element model is established.Four different hardness of rubber materials are selected.The static mechanical characteristics of the vibration isolator under various hydrostatic pressures are analysed.The static stiffness, natural frequency and transmission loss of the vibration isolators are compared with those of the isolators under atmospheric air pressure.The static stiffness of the rubber isolators under deep water increases with the hydrostatic pressure.As the hydrostatic pressure increases from 0 MPa to 10 MPa, the first order natural frequency of the rubber isolator increases by about 40%.The vibration isolator transmission loss is affected by the hydrostatic pressure.

Figure 1 .
Figure 1.Diagram of vibration isolator structure.The vibration isolator consists of a metal support bearing, a metal plate and a rubber body.For the metal support bearing, the bottom part is covered by the rubber body and the top part is connected to the equipment being isolated.The plate is the metal structure of the vibration isolator.The rubber body is the main factor to determine the vibration isolation characteristics.The vibration isolator is simplified because of its structural complexity.The rubber vibration isolator model is cut into 1/4 models along the symmetry plane.The reason for this is that the geometric model, the loads and the boundary conditions are symmetrical.The rubber material is modeled using the constitutive model chosen above.The linear elastic material model is used for the metal material.The rubber and metal cells of the isolator are meshed using solid187.The mesh is shown in figure2.There are in total 30,846 cells.Restraint and static loads are applied to the model.Different hydrostatic pressures are achieved in the form of prestress applied to the surface of the vibration isolator.

Figure 4 .
Figure 4. Vibration isolator stiffness in rubber materials.The relevant conclusions can be drawn from the above graphs.The stiffness of the vibration isolator increases with the increase in rubber hardness.The static stiffness of the vibration isolator increases significantly with an increase in hydrostatic pressure from 0Mpa to 10Mpa.This shows that the

Figure 6 .
Figure 6.Vibration isolator transmission losses at different water depths.(a) Rh=75, (b) Rh=70, (c) Rh=65 and (d) Rh=60.As seen in the above diagram, the vibration isolation effect of the vibration isolator decreases under deep water conditions.The transmission loss of the vibration isolator reduces with hydrostatic pressure.With the increase in hydrostatic pressure from 0 to 10 Mpa, the transmission loss reduces by around 4 to 7 dB.The vibration isolation effect of the isolators is affected by the different materials.As the rubber material chosen is softer, the less effect it will have.

Figure 8 .
Figure 8. Vibration isolators in different rubber materials maximum stresses.Figure8illustrates the maximum equivalent stresses of vibration isolators with different rubber materials under different water depth conditions.It increases with increasing water depth.As shown in figure7, the maximum equivalent stresses generally appear at the bottom of the vibration isolator and where it is connected to the vibration-isolated object.

Figure
Figure 8. Vibration isolators in different rubber materials maximum stresses.Figure8illustrates the maximum equivalent stresses of vibration isolators with different rubber materials under different water depth conditions.It increases with increasing water depth.As shown in figure7, the maximum equivalent stresses generally appear at the bottom of the vibration isolator and where it is connected to the vibration-isolated object.