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An integrated structure of air spring for ships and its strength characteristics

  • Yuqiang Cheng , Lin He , Changgeng Shuai , Cunguang Cai and Hua Gao EMAIL logo
Published/Copyright: August 28, 2023
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Science and Engineering of Composite Materials
From the journal Science and Engineering of Composite Materials Volume 30 Issue 1

Abstract

The increasing demand for the impact resistance of ships entails the excellent impact resistance and deformability of air spring (AS) for ships under high internal pressure. However, an AS of traditional structure is difficult to satisfy such new demand since it is designed with only 10 mm maximum deformation under high internal pressure and strong impact. For this reason, an integrated structure of filament winding reinforced bellows is proposed to ensure the structural strength at the connection of AS bellows and flanges under strong impact and high internal pressure. On this basis, a parameterized design of integrated bellows strength is explored, and we analyzed how the strength of the bellows was affected by the parameters of geometrical structure, material features, and filament winding. As proved in tests, the integrated structural design of the bellows could ensure the strength of AS under strong impact and high internal pressure, and the test result perfectly matches with the calculated strength of the integrated bellows as well as the optimized design.

Keywords: air spring; fiber-reinforced composite; strength characteristics

1 Introduction

An air spring (AS) is an excellent element for vibration isolation because of its low fixed frequency, high bearing capacity, creep-free, adjustable air pressure, and controllable performance. It has therefore been widely applied in the vibration isolation for vehicles and ships [1,2,3]. An AS for ships is normally embedded into a floating raft vibration isolation device on ships. By virtue of concentrated vibration isolation, it attenuates the vibration transmitted by large power equipments in cabins to their walls. Generally, it features higher working pressure owing to the confined cabin space for installation and the heavy load of floating raft vibration isolation device up to one hundred tons. The working pressure is often more than three times of the working pressure for an AS of the same size for vehicles [4]. In order to guarantee its reliability of service under high working pressure, an AS for ships often uses the more stable single chamber structure and its maximum deformation of no more than 10 mm. Nevertheless, floating raft vibration isolation has further developed, gradually bringing the attention to the design of a floating raft vibration isolation device with great impact resistance apart from basic vibration isolation. An AS for ships must have excellent impact resistance and deformability to guarantee the sufficient buffer of the floating raft vibration isolation device under strong impact. Therefore, its maximum deformation of 10 mm could not satisfy the needs in the impact resistance design for ships.

The overall structural strength of an AS under strong impact and high internal pressure depends on the strength at the connection between the bellows and the flanges and the structural strength of the bellows. Presently, the connection of the AS bellows with the flanges is mainly designed with a bellows vulcanized self-sealing structure, a bolt and flange extruded structure, or a bellows curled self-sealing structure. All these structures can meet the requirements for the operation of an AS under low pressure, but still fail to keep the bellows from detaching from the metal flanges when the internal pressure further increases or the bellows are severely deformed by the external impact on the AS [5,6,7]. For this reason, an integrated design of filament winding reinforced bellows is proposed to guarantee the strength of the connection with the bellows through the integrated mesh filament winding on the metal flanges. This design does not rely on the friction between the bellows and the flanges to guarantee the strength of the connection and theoretically ensures no detachment of the bellows except under the rupture of fiber. Additionally, the strength of bellows must be also significantly enhanced to make the AS for ships applicable to the extreme conditions of strong impact and high internal pressure as well as the harsh operating environment in cabins. Under normal circumstances, its burst pressure must be more than ten times of working pressure, which is much higher than the common requirement for ASs, that is, three times of working pressure in the design extreme conditions. Hence, a parameterization model for the strength of the integrated bellows must be constructed to guide the structural design of the bellows [8].

The rubber bellows of an AS are generally composed of inner and outer rubber covers and middle fiber winding layer. The fiber winding layer, as the main force carrier of the bellows, directly determines the structural strength of the bellows. It is often made of high-strength fiber and rubber through vulcanization [9]. As a fiber filament reinforced rubber composite, the bellows of an AS are typically anisotropic, making it very complicated to build its theoretical model. Presently, the theoretical studies on the mechanical properties of an AS focus on building and solving the internal air model [10,11,12]. Because of their complex mechanical characteristics, the bellows are mostly modeled by equivalent simplification, numerical simulation, or coefficient fitting. For instance, Chen et al. [13] put forward an AS stiffness calculation model, which contained a rubber bellows model. The rubber bellows model was a nonlinear equivalent simplification model formed by a fractional Kelvin–Voigt model and a smooth friction model. Zhu et al. also created an AS bellows equivalent simplification model, which took into account the rubber friction and visco-elasticity of the bellows. The key parameters of the equivalent simplification model were determined by the statistical method [14,15,16]. Qi et al. [17] constructed a parameter fitting model for the mechanical properties of the bellows through the curve fitting of numerical simulation results and carried out a prototype test to verify the effectiveness of such model. Wong et al. [18] utilized the Rebar unit in the ABAQUS software to established a finite element model for the rubber bellows to analyze how such parameters as filament winding angle and effective radius affect the mechanical properties of the AS.

The current studies on the mechanical properties of an AS focus on its stiffness, but they do not build a parameterization calculation model for the bellows. Therefore, it is difficult to apply their findings in the strength calculation and analysis of the bellows. The bellows of an AS are made of fiber filament reinforced rubber composite. The thickness of the bellows is much smaller than the curvature radius, so that the bellows of an AS can be regarded as a fiber filament reinforced composite shell structure. So far, Scholars have extensively studied the construction of a composite shell parameterization model. For instance, Ambartsumian [19] analyzed the anisotropy of fiber filament in the composite shell model for the first time and then established a theoretical model for the orthogonal anisotropic shell with stretching-bending coupling. Dong et al. [20] derived a theoretical model for the random anisotropic shell based on the theoretical model for the orthogonal anisotropic shell, which enhanced the generality of the theoretical model for fiber filament reinforced composite shells. A large number of scholars have explored the practical issues in the engineering application of composite shell models and introduced the coupling characteristics including tensile, bending, twisting, and shear into the Naghdi, Flügge, Reissner and other shell models [21,22,23,24]. Their efforts have improved the method for constructing a theoretical model for anisotropic shells in various applications. For this reason, the researching findings of shell theory can be borrowed to convert and apply the modeling and analysis methods of shell theory in the strength analysis of the integrated bellows. Presently, the studies on the strength of composite shell focus on composite hose. For instance, Gao et al. [25] carried out the theoretical analysis, simulation calculation, and prototype test to explore the strength of metal lined composite hoses. Meanwhile, they analyzed the mechanical behavior of hose fiber and lining and the stress variation at different positions of shell. Chang et al. [26] constructed a theoretical model for the material failure under progressive damage based on continuum damage mechanics and used it to calculate and analyze the strength of composite shells. Subsequently, they added the UMAT subroutine to the ABAQUS software, established, and solved the composite shell finite element model, so as to verify the correctness of the theoretical model. Based on the theoretical model for the thick shells with anisotropy, Tomasz [27] took into account the yielding and strengthening effect of metal to analyze and explore the strength of metal lined shells under internal pressure and external axial load. Shah et al. [28] used the WCM module in the ABAQUS software to construct a finite element model for fiber filament reinforced pressure vessels. With some failure models including Tsai-Hill, Tsai-Wu, and Hashin progressive damage, they calculated the strength of vessel damage and analyzed the influence of filament winding angle and winding layers on shell strength.

A composite hose can be made into a cylindrical shell model and has the same filament winding angle at any position of the shell. Differently, the integrated bellows have a variable diameter structure and also contain complicated variable winding trajectory, making it much more difficult to construct and solve the parameterization mechanical model of the integrated bellows. In this article, a composite shell theoretical model is utilized and combined with the integrated structural design and molding process of the bellows to build a mechanical model for the bellows with complicated variable winding trajectory. Moreover, the extended homogeneous capacity precision integration method is employed with the Tsai-Hill strength theory to determine the limit pressure of the integrated bellows. How design parameters affect the strength of the bellows is analyzed to explore the integrated design and strength of an AS for ships under strong impact and high internal pressure.

2 Structural design of integrated fiber filament winding reinforced bellows

An integrated filament winding reinforced structure is designed and proposed for the bellows of an AS for ships, in order to satisfy the requirement for high strength at the connection of the bellows with the metal flanges. This design allows the rubber bellows to detach from the upper and lower metal flanges under strong impact and high internal pressure. This integrated filament winding reinforced AS consists of upper mount plate, upper flange, rubber bellows, lower flange, lower mount plate, constraining sleeve, and guide seat, etc. The overall structure of the AS is shown in Figure 1 with F standing for external load, P for internal air pressure, and α for oriented angle.

Figure 1 Structure of a novel AS for ships.
Figure 1

Structure of a novel AS for ships.

The structure of the integrated filament winding reinforced bellows is illustrated in Figure 2. The integrated filament winding helps hold together the upper and lower metal flanges of the AS to improve the overall connection. In this case, the AS does not need to overcome the friction between the rubber bellows and the metal flanges when the bellows detach from the flanges under strong impact or high internal pressure. The upper and lower metal flanges are tied together by the mesh winding force of the integrated filament winding layer. The bellows cannot detach from the flanges only if the fiber is broken. Therefore, this design greatly enhances the strength of the connection between the bellows and the flanges under strong impact and high internal pressure. Moreover, the integrated filament winding reinforced bellows can be nearly fully enclosed except an air inlet is kept on the upper metal flange. However, the existing structure requires the air inlet on the upper and lower metal flanges. Hence, the proposed structure can significantly reduce the sealing area of the bellows, so as to effectively improve the overall tightness of the AS.

Figure 2 Structure of integrated filament winding reinforced bellows.
Figure 2

Structure of integrated filament winding reinforced bellows.

3 Strength model for integrated filament winding reinforced bellows

3.1 Establishing a theoretical model

As shown in Figure 1, a metal constraining sleeve is placed outside the straight part of the AS bellows. Thus, the strength of the AS bellows depends mainly on the structural strength of the arc part. The arc part of the bellows is simplified into a rotary shell structure as illustrated in Figure 3. The surface coordinates at any point of the shell are denoted by (φ, θ), where φ stands for the coordinate in the latitudinal direction and θ represents the coordinate in the longitudinal direction. The principal curvature radii are R φ and R θ , respectively. The Lame coefficients of the rotary shell are R 0 and R φ , respectively. The function of structural parameters is obtained as follows:

(1) R θ = R φ + R e / sin φ ( φ ≤ π ) R θ = − ( R φ + R e / sin φ ) ( π < φ ) R 0 = R θ sin θ .

Figure 3 Geometrical structure of the shell.
Figure 3

Geometrical structure of the shell.

The bellows of an AS are essentially made of rubber-based filament winding composite. The rubber and fiber layers are arranged and wound alternately. Hence, the physical equation of the bellows is obtained according to the composite theory as follows [29]:

(2) N φ N θ N φ θ M φ M θ M φ θ = A 11 A 12 0 0 0 0 A 12 A 22 0 0 0 0 0 0 A 66 0 0 0 0 0 0 D 11 D 12 0 0 0 0 D 12 D 22 0 0 0 0 0 0 D 66 ε φ ε θ ε φ θ χ φ χ θ χ φ θ

In equation (2), ε φ , ε θ , and ε φθ are the strain components of the bellows; x φ , x θ , and x φθ are the bending strain components of the bellows; N φ , N θ , and N φθ are the internal force components of the bellows; M φ , M θ , and M φθ are the moment components of the bellows; A ij is the tension compression stiffness; D ij is the bending stiffness. The stiffness coefficients are expressed by:

(3) A i j = Q ¯ i j h D i j = Q ¯ i j h 3 12

In equation (3), h is the fiber winding layer thickness (FWLT), and Q ¯ i j is the coefficient of the stiffness not in the principal direction of material. This parameter is expressed by [30]

(4) Q ¯ 11 = Q 11 cos 4 δ + 2 ( Q 12 + 2 Q 66 ) sin 2 δ cos 2 δ + Q 22 sin 4 δ Q ¯ 12 = ( Q 11 + Q 22 − 4 Q 66 ) sin 2 δ cos 2 δ + Q 12 ( sin 4 δ + cos 4 δ ) Q ¯ 22 = Q 11 sin 4 δ + 2 ( Q 12 + 2 Q 66 ) sin 2 δ cos 2 δ + Q 22 cos 4 δ Q ¯ 16 = ( Q 11 − Q 12 − 2 Q 66 ) sin δ cos 3 δ + ( Q 12 − Q 22 + 2 Q 66 ) sin 3 δ cos δ Q ¯ 26 = ( Q 11 − Q 12 − 2 Q 66 ) sin 3 δ cos δ + ( Q 12 − Q 22 + 2 Q 66 ) sin δ cos 3 δ Q ¯ 66 = ( Q 11 + Q 22 − 2 Q 12 − 2 Q 66 ) sin 2 δ cos 2 δ + Q 66 ( sin 4 δ + cos 4 δ ) .

In equation (4), δ is the fiber winding angle, and Q ij is the stiffness coefficient in the principal direction of material. The expression of Q ij is as follows:

(5) Q 11 = E 1 / ( 1 − υ 12 υ 21 ) , Q 12 = υ 21 E 1 / ( 1 − υ 12 υ 21 ) Q 22 = E 2 / ( 1 − υ 12 υ 21 ) , Q 66 = G 12 ,

where E 1 and E 2 stand for the elastic modulus of the fiber-reinforced composite in the principal direction 1 and 2, respectively; υ 12 and υ 21 are the Poisson’s ratio of the fiber-reinforced composite in the direction 1–2 and 2–1, respectively; G 12 is the fiber-reinforced composite shear modulus. The principal direction 1 of material is the axial direction of fiber, while the principal direction 2 of material is the lateral direction of fiber.

As illustrated in Figure 4, the integrated bellows are formed in two stages, that is, mandrel filament winding and bellows extrusion molding. The filament winding trajectory inside the bellows is also determined in such two stages. In the stage of mandrel filament winding, fiber is wound around the molded mandrel of the bellows through non-geodesic winding, which creates the initial distribution of filament winding trajectory. In the stage of bellows extrusion molding, the conical part of the bellows is extruded into the arc part of the bellows. Subsequently, a tiny change happens to the initial distribution of the filament winding trajectory, so that the final trajectory appears. According to the non-geodesic winding theory and the stable crossed fiber assumption, the variable winding trajectory equation for different positions of the bellows is created [31], and substituted into equation (4). After that, it is combined with equations (2), (3) and (5) to obtain the physical equation containing the variable winding trajectory.

Figure 4 Formation stages of integrated filament winding reinforced bellows. (a) Mandrel filament winding and (b) Bellows extrusion molding.
Figure 4

Formation stages of integrated filament winding reinforced bellows. (a) Mandrel filament winding and (b) Bellows extrusion molding.

3.2 Solving the theoretical model

The state vector η (φ) and the intermediate vector of displacement ξ(φ) are introduced to solve the mechanical model of the bellows. η (φ) is formed by eight displacement terms and internal force term, while ξ(φ) contains eight displacement terms and displacement derivative term. η (φ) and ξ(φ) include the following parameters:

(6) η ( φ ) = u ( φ ) v ( φ ) w ( φ ) κ φ ( φ ) N φ ( φ ) S φ ( φ ) V φ ( φ ) M φ ( φ ) T ξ ( φ ) = u ( φ ) v ( φ ) w ( φ ) ∂ u ( φ ) ∂ φ ∂ v ( φ ) ∂ φ ∂ w ( φ ) ∂ φ ∂ 2 w ( φ ) ∂ φ 2 ∂ 2 κ φ ( φ ) ∂ φ 2 T .

In equation (6), u, v, and w are the displacement components of the bellows; κ φ and κ θ are the curvature components of the bellows. The geometrical equation and equilibrium equation of shell [23] are combined with the physical equation containing the variable winding trajectory. After variable elimination and transformation, the function relationship expression between η (φ) and ξ(φ) is obtained with the first-order differential equation of ξ(φ) as follows:

(7) η ( φ ) = B ( φ ) ξ ( φ ) ,

(8) d ξ ( φ ) d φ = C ( φ ) ξ ( φ ) + q ( φ ) .

In equations (7) and (8), B (φ) is the eighth-order transformation matrix; C (φ) is the eighth-order coefficient matrix; and q (φ) is the external force term. The elements of B (φ), C (φ), and q (φ) are detailed in the Appendix. The extended homogeneous capacity precision integration method is adopted to solve the non-homogeneous variable coefficient differential equation (8). Thus, the transfer relationship between the intermediate vectors of displacement at different nodes of the bellows is obtained as follows [32]:

(9) d d φ ξ ( φ ) 1 = C ( φ ) q ( φ ) 0 0 ξ ( φ ) 1 ⇒ d d φ ξ ¯ ( φ ) = C ¯ ( φ ) ξ ¯ ( φ ) ξ ¯ ( φ i + 1 ) = exp ( C ¯ ( φ i ) ⋅ ( φ i + 1 − φ i ) ) ξ ¯ ( φ i ) = T ¯ i ξ ¯ ( φ i ) .

where C ¯ ( φ ) is the expansion coefficient matrix, ξ ¯ ( φ ) is the expansion displacement vector, and T ¯ is the transfer matrix between the expansion vectors of displacement.

During the strength test of the bellows of an AS, the upper and lower mount plates are fixed, so that the boundary conditions at both ends of the bellows are:

(10) u = 0 , w = 0 , v = 0 , κ = 0 .

With equation (9) and the boundary conditions (10), all the intermediate vector of displacement at the nodes of the bellows can be solved. After being transformed with equation (7), the state vector at different positions can be obtained. The state vector is substituted into the physical equation (2) to determine the strain component in the principal direction of the bellows. Based on the stress rotation axis formula and the stress-strain relationship, the stress component in the principal direction of fiber-reinforced composite is determined as [24]:

(11) σ 1 σ 2 σ 12 = cos 2 δ sin 2 δ 2 sin δ cos δ sin 2 δ cos 2 δ − 2sin δ cos δ − sin δ cos δ sin δ cos δ cos 2 δ − sin 2 δ × Q ¯ 11 Q ¯ 12 Q ¯ 16 Q ¯ 12 Q ¯ 22 Q ¯ 26 Q ¯ 16 Q ¯ 26 Q ¯ 66 ε φ ε θ ε φ θ .

As for fiber composite, Tsai-Hill strength theory can be adopted to judge the failure. The Tsai-Hill failure criterion is

(12) σ 1 2 X 2 − σ 1 σ 2 X 2 + σ 2 2 Y 2 + τ 12 2 S 2 = 1 .

In equation (12), X, Y, and S represent the axial, lateral, and shear tensile strengths of fiber composite, respectively. The stresses σ 1, σ 2, and σ 12 in the principal direction of fiber composite are obtained by equation (11) and then substituted into equation (12) to judge the failure of the integrated bellows.

4 Factors affecting the strength of integrated filament winding reinforced bellows

The strength design of the AS bellows involves three types of parameter, that is, geometrical structure parameters including effective radius R φ , bellow radius R e , and oriented angle α; material characteristics parameters including tensile strengths in three directions of fiber composite X, Y, Z, and FWLT h; and filament winding parameters including initial angle γ and slip coefficient λ.

Burst pressure refers to the limit pressure that an AS can withstand under constant pressurization before bursting. The structural design of the integrated filament winding reinforced bellows has greatly enhanced the strength at the connection of the bellows with the flanges, so that the burst pressure of an AS can reflect the structural strength of the integrated bellows. An M-15T AS was taken as an example to analyze the influence of different design parameters on the strength of its bellows. The burst pressure was calculated by altering a design parameter. Therefore, the design parameters of the strength for the bellows of the AS were normalized to simplify the analysis of the design parameters and identify the key factors affecting the strength of the bellows. The normalized parameters are expressed by:

(13) R φ ′ = R φ / R φ 0 , R e ′ = R e / R e 0 , h ′ = h / h 0 , X ′ = X / X 0 , Y ′ = Y / Y 0 , Z ′ = Z / Z 0 .

In equation (13), X ′ , Y ′ , Z ′ , R φ ′ , R e ′ and h ′ are the normalized values of the design parameters; R φ 0 , R e 0 , h 0 , X 0 , Y 0 , and Z 0 are the parameters of the M-15T AS prototype, and they are detailed in Table 1.

Table 1

Parameters of M-15T AS prototype

Parameter Value Parameter Value Parameter Value
Bellow radius R φ 0 22.5 mm Lateral tensile strength Y 0 4.7 MPa [33] Shear modulus G 12 2.5 MPa [34]
Effective radius R e 0 141 mm Shear tensile strength Z 0 4.0 MPa [33] Oriented angle β 54°
Thickness of filament winding layer h 0 4.4 mm Elastic modulus E 1 49.7 GPa [34] Initial winding angle γ 24.8°
Axial tensile strength X 0 347.6 MPa [33] Elastic modulus E 2 6 MPa [34] Slip coefficient λ 0.2

4.1 Influence of geometrical structure parameters on the strength of bellows

The burst pressure of the AS is calculated using different effective radii, bellow radii, and oriented angles. The results are detailed in Figure 5. As shown in Figure 5(a), the strength of the bellows decreases with the increased effective radius and bellow radius, and such decrease is lower when the parameter is larger. As revealed in the comparison, the bellow radius affects the strength of the bellows more significantly than the effective radius. In addition, the model of an AS is generally determined by its rated load capacity, which is directly influenced by the effective radius. Therefore, once the AS model is determined, it is generally not optimized by adjusting the effective radius to improve its strength characteristics. As illustrated in Figure 5(b), the strength of the bellows decreases with the increase of oriented angle α, and such decrease is lower when the oriented angle is greater. When α = 0°, the AS has the largest strength of the bellows. The burst pressure changes by only 7% of its initial value within the variation range of then oriented angle. Compared with effective radius and bellow radius, the influence of the oriented angle on the strength of the bellows is basically ignorable.

Figure 5 Variation curves of burst pressure with the geometrical structure parameters of an AS: (a) variation curve of burst pressure with effective radius and bellow radius; (b) variation curve of burst pressure, with oriented angle.
Figure 5

Variation curves of burst pressure with the geometrical structure parameters of an AS: (a) variation curve of burst pressure with effective radius and bellow radius; (b) variation curve of burst pressure, with oriented angle.

4.2 Influence of material characteristics parameters on the strength of bellows

The burst pressures of the AS with different axial tensile strengths X′, lateral tensile strengths Y′, shear strengths S′, and FWLT h′ are calculated as shown in Figure 6. As illustrated, the variation of X′ and Y′ has no influence on the strength of the bellows. However, the increase of Y′ and h′ makes a clear contribution to the enhanced strength of the bellows. Evidently, the damage to the bellows of the AS is mainly attributed to the insufficient lateral tensile strength of the material when the bellows are affected by the lateral stress component. Therefore, the strength of the bellows could be enhanced by improving the lateral tensile strength of the material or lowering the lateral stress component with the increased FWLT.

Figure 6 Variation curves of burst pressure with the material characteristics parameters.
Figure 6

Variation curves of burst pressure with the material characteristics parameters.

As shown in Figure 6, the influence of the lateral tensile strength on the strength of the bellows is basically equal to that of the FWLT when the normalized value varies below 1.2. With the increase of the normalized value, their contribution to the enhanced strength of the bellows diminishes. When the normalized value is within [1.2, 3.4], the lateral tensile strength exerts a stronger effect on the strength of the bellows than the FWLT. However, the further increase of the normalized value makes the FWLT a bigger contributor to the strength of the bellows than the lateral tensile strength. In this case, the lateral tensile strength is not a dominant constraint on the strength of the bellows anymore. In the practical fabrication of bellows, the structure and size of the mold must be adjusted to any changed FWLT. Therefore, if the overall structure of the integrated bellows is fixed, a material with higher lateral tensile strength can be selected to effectively enhance the strength of the bellows.

4.3 Influence of filament winding parameters on the strength of bellows

In the practical fabrication of the integrated bellows, any technically feasible filament winding trajectory must fit the non-geodesic winding model. Hence, the internal filament winding trajectory of the bellows would be determined by the initial filament winding angle (IFWA) γ and the slip coefficient λ in the non-geodesic winding model. Meanwhile, the maximum value of the IFWA and the slip coefficient must be limited to guarantee the realizable winding. The maximum value of the IFWA is not greater than 35°, since the winding cannot be realized if the filament winding angle exceeds 90° at any point in the process of winding. The maximum value of the slip coefficient should not exceed 0.25. If exceeding 0.25, the winding will fail since fiber might slip because of insufficient friction on the mandrel surface in the process of winding.

In order to analyze the influence of the filament winding parameters on the strength of the bellows, the burst pressure of the AS is calculated with the limited maximums of the IFWA and the slip coefficient. As shown in Figure 7, the strength of the bellows increases and then decreases with the increasing the IFWA. There is a maximum strength. When the IFWA is low, the slip coefficient has a very little influence on the strength of the bellows. With the increase of the IFWA, the influence of the slip coefficient on the strength of the bellows becomes stronger gradually. Moreover, the IFWA for the maximum strength of the bellows becomes larger with the increase of the slip coefficient. Therefore, the slip coefficient should be enhanced as much as technically feasible in the process of mandrel winding, so as to improve the strength of the bellows. After that, the suitable IFWA is determined by the slip coefficient.

Figure 7 Variation curves of burst pressure with the filament winding parameters.
Figure 7

Variation curves of burst pressure with the filament winding parameters.

5 Prototype fabrication and strength test of a novel AS for ships

5.1 Prototype fabrication of an AS

The burst test and impact test are intended to verify the reliability of the connection between the integrated bellows and flanges of the AS, and the effectiveness of the theoretical model for the strength of the bellows. The M-8T, M-15T, and M-30T AS prototypes were fabricated to test the strength of AS under strong impact and high internal pressure. The prototype parameters of such three ASs are detailed in Table 2.

Table 2

Parameters of three AS prototypes

M-8T AS M-15T AS M-30T AS
Parameter Value Parameter Value Parameter Value
R φ 26 mm R φ 22.5 mm R φ 32.5 mm
R e 108 mm R e 141 mm R e 195 mm
α 90° α 54° α 60°
h 6.6 mm h 4.4 mm h 8.8 mm
γ 28.9° γ 24.8° γ 27.2°
λ 0.25 λ 0.2 λ 0.25

The prototypes of the integrated filament winding reinforced AS were fabricated in four steps. First, the mandrel structure was determined with the design parameters of an AS prototype. Second, a three-dimensional model of the molded mandrel was drafted, and the filament winding parameters were uploaded into the CADWIND software for the model of integrated filament winding. Third, numerical control codes were used to complete the integrated filament winding program, which was input into the filament winding machine for the integrated filament winding of mandrel. Fourth, the integrated bellows were placed into the mold after winding, and then formed by extrusion and vulcanization. They were subsequently assembled into an AS prototype. The numerical modeling and actual process of the integrated filament winding are illustrated in Figure 8.

Figure 8 Fabrication process of integrated filament winding reinforced bellows.
Figure 8

Fabrication process of integrated filament winding reinforced bellows.

5.2 Burst test of an AS

In the burst test, an AS was arranged at its rated height in the test clamping apparatus. The 4DSY-80 electric pressure pump was used to inject water into the AS. The water injection rates of the pump vary at different pressures: it is 432 L/h at low pressure (≤1.6 MPa) and 20 L/h at high pressure (>1.6 MPa). When the bellows ruptures, the reading of the pressure gauge was recorded, and took as the burst pressure of the AS. After the burst test, the AS can be removed from the test fixture. Upon dismantling the lower mount plate, the damage to the bellows can be observed. The installation and burst test of the AS are shown in Figure 9.

Figure 9 Installation and burst test of an AS.
Figure 9

Installation and burst test of an AS.

The burst pressures obtained from the burst test of three AS prototypes at the rated height are given in Table 3. The deviation from the theoretical calculation is within 10%. The theoretical results are basically consistent with the test results. This proves the effectiveness of the theoretical model for the strength of the integrated bellows. Additionally, the burst pressure in the test exceeded 10 times of working pressure for all these ASs. The damage happened to their bellows, but no bellows detached from the flanges. This verifies the reliability of the connection in the integrated bellows under high internal pressure.

Table 3

Burst test results of Ass

Prototype Rated air pressure (MPa) Calculated compressive strength (MPa) Tested compressive strength (MPa) Calculation error (%) Safety coefficient
M-8T 2.0 22.3 21.0 6.2 10.5
M-15T 2.3 28.9 27.0 7.0 11.7
M-30T 2.4 37.3 35.5 5.1 14.8

When the overall structural parameters and formation process of an AS are certain, it is most efficient and economical to optimize the filament winding trajectory in the bellows. As illustrated in Figure 7, the M-15T AS had the largest strength of the bellows, up to 36.5 MPa, when the IFWA was 33.8° and the slip coefficient is 0.25. The M-15T AS was fabricated with the optimal filament winding parameters, and then put into a burst test. After optimizing the prototype design, the burst pressure was enhanced from 27.0 to 34.0 MPa, and the strength of the bellows was improved by 25.9% with a theoretical calculation error of 7.4%. The theoretical results were basically consistent with the test results. This further proves the correctness of the parameterization model for the strength of the integrated bellows.

5.3 Impact test of an AS

The structure and application of the impact test equipment are as shown in Figure 10. Before the impact test, an AS was arranged at the rated height in the test clamping apparatus. Air was pumped into it till the rated pressure. During the impact test, the AS was fixed with the test clamping apparatus onto a five-ton drop hammer. The hammer fell freely to impact the AS prototype. An acceleration sensor was used to collect the data of the impact test, and the impact force and displacement were calculated on this basis. After test, the overall appearance of the AS and the damage to the bellows were examined.

Figure 10 Structure and application of impact test equipment. 1 – impact base; 2 – impact head; 3 – mount frame; 4 – tested prototype; 5 – guide plate; 6 – impact hammer; 7 – acceleration sensor; 8 – release unit.
Figure 10

Structure and application of impact test equipment. 1 – impact base; 2 – impact head; 3 – mount frame; 4 – tested prototype; 5 – guide plate; 6 – impact hammer; 7 – acceleration sensor; 8 – release unit.

Different working scenarios require different impact performance of ASs. M-8T and M-30T ASs are only required to achieve an impact deformation capacity of about 15 mm, and to ensure the impact stability of the application, M-8T and M-30T ASs are required to have internal limiters so that the deformation cannot be increased after the deformation reaches 15 mm. The application scenario for the M-15T AS is the most demanding. It requires a large impact deformation capability of at least 50 mm. The purpose of this article is to provide a novel bellows structure that greatly increases the impact deformation capacity of the AS bellows under high pressure conditions. Since the maximum impact deformation capacity of conventional AS bellows is only 10 mm, in order to reflect the impact large deformation capacity of the novel bellows structure, this article provides the impact test results of the M-15T type AS with the impact characteristic curves shown in Figure 11. During the impact test, the AS had the maximum impact deformation of 50.8 mm. After the impact test, the AS kept intact on the whole, and the bellows were not damaged. This proves the reliable connection of the integrated filament winding reinforced bellows under strong impact and dramatic deformation.

Figure 11 Impact characteristic curve of an AS.
Figure 11

Impact characteristic curve of an AS.

6 Conclusion

In this article, integrated filament winding reinforced bellows are proposed to meet the requirements for high strength at the connection of the bellows with the flanges when the AS for new ships is exposed to strong impact and high internal pressure. Moreover, theoretical modeling, parameter analysis, and design optimization are carried out with regard to the strength of the integrated bellows. Some conclusions are mainly drawn as follows:

  1. The structural design of integrated filament winding reinforced bellows is put forward to construct a theoretical model for the strength of the integrated bellows with variable winding trajectory. The prototypes of the integrated filament winding reinforced AS are fabricated and tested for strength. As revealed in the test results, the bellows of the AS are not detached from the flanges when the AS is tested with a deformation of more than 50 mm under strong impact, or with a high internal pressure exceeding 10 times of working pressure. This demonstrates the reliable connection of the integrated bellows under strong impact and high internal pressure. Moreover, the theoretical calculation error of burst pressure is within 10% for three AS prototypes and the optimized design prototype, which verifies the effectiveness of the parameterization calculation model for the strength of the integrated bellows.

  2. The strength of the integrated bellows is mainly affected by geometrical structure parameters, material characteristic parameters, and filament winding parameters. Among the geometrical structure parameters, the influence of the oriented angle on the strength of the bellows is basically ignorable. The strength of the bellows decreases with the increase of the effective radius and the bellow radius. The bellow radius exerts a stronger effect on the strength of the bellows. Among the material characteristic parameters, improving the lateral tensile strength and shear strength has barely affected the strength of the bellows. The damage to the bellows must be mainly attributed to the insufficient lateral tensile strength of the material when the bellows are affected by the lateral stress component. Therefore, the strength of the bellows can be noticeably improved by enhancing the lateral tensile strength of the material or increasing the FWLT in the bellows. As for the filament winding parameters, the strength of the bellows goes up and down with the variation of the IFWA, and reaches a maximum level. The maximum strength becomes larger with the increase of the slip coefficient.

  1. Funding information: The authors wish to acknowledge, with thanks, the financial support from the Naval University of Engineering.

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Appendix

Coefficient matrix elements

  1. Elements of the transformation matrix B (φ)

    B ( φ ) = 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 B 4 , 1 0 0 0 B 4 , 5 0 0 0 B 5 , 1 0 B 5 , 3 B 5 , 4 0 0 0 0 B 6 , 1 B 6 , 2 0 B 6 , 5 0 0 0 0 B 7 , 1 0 0 B 7 , 4 0 B 7 , 6 0 B 7 , 8 B 8 , 1 0 0 B 8 , 4 0 B 8 , 6 B 8 , 7 0

    B 4 , 1 = 1 R φ , B 4 , 5 = − 1 R φ , B 5 , 1 = A 12 cos φ R 0 , B 5 , 3 = A 11 R φ + A 12 sin φ R 0 φ , B 5 , 4 = A 11 R φ , B 6 , 1 = − A 66 R 0 − D 66 sin φ R 0 2 R φ , B 6 , 2 = − A 66 R 0 ′ R 0 R φ − 2 D 66 sin 2 φ R 0 ′ R 0 3 R φ + D 66 sin φ cos φ R 0 2 R φ , B 6 , 5 = A 66 R φ + D 66 sin 2 φ R 0 2 R φ , B 7 , 1 = − D 12 sin φ R 0 R φ 2 − cos 2 φ D 22 R 0 2 R φ , B 7 , 4 = R 0 ′ D 11 R 0 R φ 3 , B 7 , 6 = D 12 sin φ R 0 R φ 2 , B 7 , 8 = D 11 R φ 2 , B 8 , 1 = D 12 cos φ R 0 R φ , B 8 , 4 = D 11 R φ 2 , B 8 , 6 = − D 12 cos φ R 0 R φ , B 8 , 7 = − D 11 R φ 2

    A ij and D ij are the stiffness coefficients of composite [31].

  2. Elements of the coefficient matrix C (φ)

    C ( φ ) = 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 C 4 , 1 0 C 4 , 3 C 4 , 4 0 C 4 , 6 C 4 , 7 C 4 , 8 0 C 5 , 2 0 0 C 5 , 5 0 0 0 0 0 0 0 0 0 1 0 C 4 , 1 0 C 4 , 3 C 4 , 4 0 C 4 , 6 C 4 , 7 C 4 , 8 − R φ C 8 , 1 0 C 8 , 3 C 8 , 4 0 C 8 , 6 C 8 , 7 C 8 , 8

    C 4 , 1 = sin φ ( A 12 R φ 2 + D 12 ) R 0 + R φ ( A 22 R φ 2 + D 22 ) cos 2 φ R φ A 11 R 0 2

    C 4 , 3 = − A 11 R 0 2 R 0 ′ + A 22 R φ R 0 R φ cos φ sin φ A 11 R 0 3

    C 4 , 4 = − R ′ ( A 11 R φ 2 + D 11 ) R 0 R φ 2 A 11

    C 4 , 6 = − R φ A 11 R 0 2 + ( ( − A 12 R φ 2 − D 12 ) sin φ + P R φ 2 R 0 ) R 0 − R φ D 22 cos 2 φ R φ A 11 R 0 2

    C 4 , 7 = D 11 R 0 ′ R 0 R φ 2 A 11

    C 4 , 8 = − D 11 A 11 R φ

    C 5 , 2 = R 0 ( A 66 R 0 2 − 2 D 66 cos 2 φ + 2 D 66 ) R 0 ″ + cos φ R φ R 0 2 A 66 − 2 R φ cos 2 φ D 66 + 2 sin φ D 66 R 0 + 2 R φ D 66 R 0 ′ + ( 2 D 66 cos 2 φ − 2 D 66 ) R 0 ′ 2 + ( − D 66 cos 2 φ + D 66 ) R 0 2 − D 66 R 0 cos 2 φ sin φ R φ R 0 2 ( A 66 R 0 2 − D 66 cos 2 φ + D 66 ) C 5 , 5 = − A 66 R 0 2 cos φ R φ + D 66 R φ cos 3 φ − 2 D 66 R 0 sin φ cos φ − 2 D 66 R 0 ′ cos 2 φ − D 66 R φ cos φ + 2 D 66 R 0 ′ R 0 ( A 66 R 0 2 − D 66 cos 2 φ + D 66 ) C 8 , 1 = ( ( ( A 12 R φ 2 + D 12 ) R θ + R φ 3 A 22 ) R 0 2 + R θ R φ 2 D 22 sin φ R 0 ) cos φ − R θ R φ R 0 ′ D 22 cos 2 φ D 11 R 0 3 R θ C 8 , 3 = ( ( A 11 R θ + A 12 R φ ) R 0 2 + R φ ( ( A 12 R θ + A 22 R φ ) sin φ − P R θ R φ ) R 0 ) R φ D 11 R 0 2 R θ C 5 , 5 = − A 66 R 0 2 cos φ R φ + D 66 R φ cos 3 φ − 2 D 66 R 0 sin φ cos φ − 2 D 66 R 0 ′ cos 2 φ − D 66 R φ cos φ + 2 D 66 R 0 ′ R 0 ( A 66 R 0 2 − D 66 cos 2 φ + D 66 )

    C 8 , 4 = − R 0 ″ D 11 R θ R 0 + ( R φ ( A 11 R θ + A 12 R φ ) R 0 2 + R θ D 12 R 0 sin φ + R θ R φ D 22 cos 2 φ − P R φ 3 R θ R 0 ) R φ R φ D 11 R θ R 0 2

    C 8 , 6 = D 22 cos 2 φ R φ R 0 ′ − D 12 cos φ R 0 2 + 2 R 0 R φ D 22 cos φ sin φ D 11 R 0 3

    C 8 , 7 = R 0 ″ D 11 R 0 − R φ ( D 12 sin φ R 0 + D 22 R φ cos 2 φ ) R φ D 11 R 0 2

    C 8 , 8 = − 2 R 0 ′ R 0

  3. Elements of the external force term q (φ)

q ( φ ) = [ 0 0 0 C 4 , 9 q φ C 5 , 9 q θ 0 C 4 , 9 q φ C 8 , 9 q z ]

C 4 , 9 = − R 0 3 R φ 4 D 11 R 0 3 + A 11 R 0 3 R φ 2 ⋅ 1 − D 11 R 0 3 D 11 R 0 3 + A 11 R 0 3 R φ 2 − 1

C 5 , 9 = − R 0 4 R φ 2 D 66 R 0 2 sin 2 φ + A 66 R 0 4

C 8 , 9 = − R φ 3 D 11

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Received: 2023-05-20
Revised: 2023-06-29
Accepted: 2023-07-23
Published Online: 2023-08-28

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/secm-2022-0221
Keywords for this article
air spring; fiber-reinforced composite; strength characteristics
Creative Commons
BY 4.0

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