The influence of membrane air spring cord parameters on the vertical stiffness

1. Introduction

Air springs are widely employed as crucial vibration-damping components in vehicle suspension systems and air-floating vibration isolation platforms due to their advantages, including adjustable vertical stiffness, low natural frequency, and nonlinear damping characteristics [1]. In large-load vibration isolation platforms used in high-end equipment such as carrier rockets, the installation space is often restricted. Under such limitations, the vertical stiffness of conventional air springs often fails to meet the performance requirements of large-load vibration isolation systems. Therefore, it is of practical significance to study the structural characteristics and parameter variations of rubber airbags in air springs to improve their vertical stiffness in confined spaces.

In recent years, researchers have conducted numerous studies on the vertical stiffness of air springs from the perspectives of cord parameters and other structural factors. Regarding the effect of cord parameters on vertical stiffness, Zhou, et al. [2] established a finite element model of a membrane air spring based on its nonlinear material properties, geometric configuration, and contact behavior. The results indicated that, within a certain range, cord parameters are key factors influencing the vertical stiffness characteristics of membrane air springs. Lee and Kim [3] developed a relationship between the laying angle of the cord layer and its radial position by analyzing deformation patterns under different load conditions, including vertical, lateral, and torsional loads. By comparing different layup angles, they examined how these variations affect the vertical, lateral, and torsional stiffness of the air spring. Zhang, et al. [4] discussed the nonlinear behavior of air springs from the perspectives of capsule material properties and the geometric configuration of rubber airbags. A sensitivity analysis of bladder-type air springs revealed that the number of cord layers exerts the most significant influence on vertical stiffness. To further investigate the aging mechanism of air springs, Lu, et al. [5] analyzed the effects of aging time on the vertical static stiffness of air springs. The study demonstrated that the vertical static stiffness increases with aging time, and that the aging of both the air spring cord material and rubber material contributes to this increase.

In addition, several researchers have examined the vertical stiffness of air springs from the perspective of innovative structural designs. Yin, et al. [6] proposed an air spring-based dynamic vibration absorber and investigated its damping and vibration reduction performance. The results showed that, by properly adjusting the air pressure and the throttle opening, the air spring vibration absorber could achieve a suitable vibration absorption frequency and damping ratio, thereby obtaining a broad vibration reduction effect. Oman and Nagode [7] analyzed two types of piston structures and compared their load–deformation characteristics and fatigue life. Their findings showed that the simulated and experimental vertical dynamic stiffness results were consistent with theoretical predictions. Marini, et al. [8] studied an air spring system consisting of a primary pneumatic chamber and multiple auxiliary air storage tanks, where airflow between the chambers was controlled by an adjustable valve. This configuration allowed for regulation of the equivalent spring stiffness by altering the airflow distribution.

The studies conducted by these researchers can generally be divided into two categories of air spring types: bladder air springs and membrane air springs. Between these, membrane air springs exhibit smaller variations in vertical stiffness and smoother elastic characteristic curves, which make them more suitable for applications requiring stable stiffness performance.

To further analyze the influence of rubber airbag cord parameters on the vertical stiffness of membrane air springs within restricted installation spaces, parameters such as cord spacing, cord angle, cord cross-sectional area, and the number of cord layers were selected for investigation. The orthogonal experimental method was adopted to analyze the variation patterns of vertical stiffness under two different compression displacements, thereby revealing the influence laws of these four cord parameters on the vertical stiffness behavior of membrane air springs.

2. Membrane air spring structure and vertical stiffness

A specific type of membrane air spring was selected as the research object. The outer contour height of the membrane air spring is 347 mm, the radius around the rotation axis is 144 mm, and its mass is 13.214 kg. The distance between two adjacent installation holes is 158.8 mm, the maximum working pressure is 5.0 bar, and the stroke is 431.0 mm.

The membrane air spring was modeled using SolidWorks software. The upper cover plate, rubber airbag, and bottom piston were respectively modeled as shown in Figs. 1(a), (b), and (c). Fig. 1(d) presents the three-dimensional assembly drawing.

Fig. 1Composition and overall structure of the membrane air spring

a) Upper cover plate

b) Rubber airbag

c) Bottom piston

d) Three-dimensional assembly drawing

The nonlinearity of the vertical stiffness of the membrane air spring mainly arises from the deformation stiffness of the rubber airbag, as shown in Fig. 1(b) [9]. During compression, the stiffness is influenced by changes in the effective area of the compressed air column and the deformation characteristics of the rubber airbag itself. The total stiffness can be expressed by Eq. (1):

where Eq. (1), $K_t$ represents the total vertical stiffness of the membrane air spring, $K_1$ denotes the vertical stiffness of the compressed air, and $K_2$ is the deformation stiffness of the rubber airbag.

The variation of $K_1$, representing the vertical stiffness of the compressed air, can be divided into two stages: a linear stage and a nonlinear stage. In the linear stage, $K_1$ assumes that the gas inside the rubber airbag behaves as an ideal gas. When the compression displacement of the membrane air spring is small, the change in gas state within the rubber airbag is linear and can be regarded as an isothermal process. Therefore, the pressure ($P$) and volume ($V$) of the gas satisfy Boyle’s Law [10]:

where Eq. (2), $F$ is the load-bearing capacity of the membrane air spring, $P_a$ is the internal pressure of the rubber airbag under compression, and $A$ is the effective internal area of the rubber airbag. In Eq. (3), $P_0$ denotes the gas pressure of the rubber airbag in its initial state, $V_0$ is the initial volume of the rubber airbag, $V_a$ represents the compressed volume of the rubber airbag, and $C$ is a constant. In Eq. (4), $z$ is the vertical displacement of the membrane air spring.

Based on Eqs. (2-4), the load-bearing capacity $F$ of the membrane air spring can be expressed as Eq. (5):

By differentiating $F$ with respect to $z$ according to Eq. (5), the vertical stiffness of the compressed air $K_1$ can be expressed as Eq. (6):

As the compression displacement increases further, the state of the gas within the rubber airbag gradually deviates from the linear relationship. During this nonlinear stage, the gas behavior follows a more general variable process equation [11], expressed as Eq. (7):

where Eq. (7), $\lambda$ represents the gas process coefficient.

It follows from Eq. (7) that the vertical stiffness $K_1$ of the membrane air spring is related to the initial gas pressure $P_0$ of the rubber airbag, its initial volume $V_0$, the effective area $A$ inside the airbag, and the vertical displacement $z$ of the membrane air spring.

The deformation stiffness $K_2$ of the rubber airbag is composed of the stiffness $K_c$ generated by the cord component and the stiffness $K_r$ generated by the pure rubber component, as expressed by Eq. (8). Due to the complexity of the structure and the absence of a unified theoretical expression, it is necessary to analyze four key cord parameters-number of cord layers ($t$), cord spacing ($m$), cord angle ($\alpha$), and cross-sectional area of the cord ($s$) – for the specific design:

Since the effect of the pure rubber material on $K_r$ is minimal and can be neglected, this study focuses on investigating the effects of the number of cord layers, cord spacing, cord angle, and cord cross-sectional area on $K_c$.

3. Finite element simulation of membrane air spring

3.2. Structure of the rubber airbag cord layer

The cord layer plays a decisive role in determining the vertical stiffness and durability of membrane air springs [12]. To accurately describe the structural configuration and parameters of the rubber airbag cord layer, the local capsule skin shown in Fig. 1(b) was sectioned, as illustrated in Fig. 2. The local capsule skin was magnified tenfold, and the cross-section A-A perpendicular to the cord direction was selected to construct the sectional view of the cut capsule skin. Fig. 2 presents the schematic cross-sectional view for a curtain layer comprising two layers.

Fig. 2Schematic diagram of the cord layer structure of the rubber airbag

In Fig. 2, $\alpha$ denotes the cord angle, which is the angle between the projection of a single cord’s direction and the rotation center axis. $d$ represents the diameter of a single cord’s cross-section, while $s$ is the cross-sectional area of the cord, defined as $s=\frac{\pi }{4}{d}^2$. $m$ denotes the cord spacing, referring to the distance between the central axes of two adjacent cords. $n$ represents the spacing between the axes of two adjacent cord layers, and $L$ indicates the thickness of the rubber airbag wall of the membrane air spring.

3.3. Definition of the fluid cavity

To simulate and analyze the gas flow behavior within the rubber airbag during operation, it was necessary to define the reference point coordinates of the fluid elements and the fluid cavity. The membrane air spring model created in SolidWorks was imported into ABAQUS software, where the F3D4 fluid element was selected for the internal air cavity. The F3D4 element is a five-node pyramid element, with its apex node serving as the reference point of the fluid cavity and the four base nodes corresponding to nodes on the inner surface of the cavity. A Cartesian coordinate system was established, with the origin O (0, 0, 0) positioned at the top of the central axis of the upper cover plate. The reference point O′ of the rubber airbag’s fluid cavity was defined at ($x$, $y$, $z$) = (0, −10, 0), as shown in Fig. 3.

Fig. 3Reference point of the fluid cavity

Fig. 4Mesh model of the membrane air spring

4. Analysis of the influence of cord layer parameters on the vertical stiffness of membrane air spring

4.2. Influence of cord layer parameters on the vertical stiffness of the membrane air spring

First, the variations in the vertical stiffness of the air spring are analyzed with respect to changes in each of the four cord parameters. Subsequently, the cord layer spacing, cord angles, and cord cross-sectional area are individually varied according to the values specified in Table 1, while all of the other parameters are held constant. The corresponding calculated results for vertical stiffness are presented in Table 2.

Table 2Calculated results for vertical stiffness

	$t$-$m$-$\alpha$-$s$	$K_1$ (N∙mm−1)
Cord layer spacing	2-3-15-0.283	51.413
	2-5-15-0.283	42.235
	2-7-15-0.283	35.525
	4-3-15-0.283	103.748
	4-5-15-0.283	88.791
	4-7-15-0.283	78.652
	6-3-15-0.283	124.443
	6-5-15-0.283	110.875
	6-7-15-0.283	101.487
Cord angles	2-3-5-0.100	47.575
	2-3-10-0.100	40.875
	2-3-15-0.100	31.532
	4-3-5-0.100	84.811
	4-3-10-0.100	80.625
	4-3-15-0.100	72.775
	6-3-5-0.100	105.375
	6-3-10-0.100	102.915
	6-3-15-0.100	95.975
Cord cross-sectional area	2-3-15-0.100	31.476
	2-3-15-0.166	41.775
	2-3-15-0.283	51.393
	4-3-15-0.100	72.775
	4-3-15-0.166	88.148
	4-3-15-0.283	103.751
	6-3-15-0.100	95.975
	6-3-15-0.166	110.275
	6-3-15-0.283	124.443

Thereafter, using the aforementioned three cord parameters as horizontal axes and vertical stiffness as the vertical axis, graphical representations were generated for air springs with 2, 4, and 6 cord layers, as illustrated in Fig. 5.

According to Table 2 and Fig. 5(a), when the number of cord layers remained constant, the vertical stiffness of the membrane air spring decreased with increasing cord spacing, indicating a negative correlation. For cord layers of 2, 4, and 6, the vertical stiffness decreased by 15.888 N/mm, 25.096 N/mm, and 22.956 N/mm, respectively, as the cord spacing increased from 3 to 7 mm. The corresponding reduction ratios were 30.9 %, 24.2 %, and 18.4 %, respectively. These results indicate that as the number of cord layers increased, the effect of larger cord spacing on reducing vertical stiffness became progressively weaker.

Fig. 5Vertical stiffness characteristics of membrane air springs with different cord layer parameters

a) 3 Cord spacings

b) 3 Cord angles

c) 3 Cord cross-sectional areas

When the cord spacing remained constant, vertical stiffness increased with the number of cord layers. For cord spacings of 3, 5, and 7 mm, the increase in vertical stiffness as the number of cord layers rose from 2 to 6 was 73.030 N/mm, 68.640 N/mm, and 65.962 N/mm, respectively. The corresponding increase ratios were 142.0 %, 162.5 %, and 185.7 %, respectively. When the cord spacing was small, the number of cord layers exerted a pronounced effect on the value of vertical stiffness, but a comparatively weaker effect on its rate of change. Conversely, as cord spacing increased, the number of cord layers had a smaller effect on the stiffness value but a greater influence on its rate of change. The minimum vertical stiffness of 35.525 N/mm occurred with 2 cord layers and a spacing of 7 mm, while the maximum vertical stiffness of 124.443 N/mm was obtained with 6 cord layers and a spacing of 3 mm.

According to Table 2 and Fig. 5(b), when the number of cord layers was constant, the vertical stiffness of the membrane air spring decreased with increasing cord angle, demonstrating a negative correlation. For 2, 4, and 6 cord layers, the reductions in vertical stiffness as the cord angle increased from 5° to 15° were 16.043 N/mm, 12.036 N/mm, and 9.400 N/mm, respectively, corresponding to decrease ratios of 33.7 %, 14.2 %, and 8.9 %. These results indicate that as the number of cord layers increases, the influence of a larger cord angle on reducing vertical stiffness weakens progressively.

When the cord angle remained constant, the vertical stiffness increased with the number of cord layers. For cord angles of 5°, 10°, and 15°, as the number of cord layers increased from 2 to 6, the vertical stiffness increased by 57.800 N/mm, 62.040 N/mm, and 64.443 N/mm, respectively, corresponding to increase ratios of 121.5 %, 151.8 %, and 204.4 %. This indicates that as the number of cord layers increased, the impact of the cord angle on the enhancement of vertical stiffness became more pronounced. The minimum vertical stiffness of 31.532 N/mm occurred when the number of cord layers was 2 and the cord angle was 15°, while the maximum vertical stiffness of 105.375 N/mm occurred when the number of cord layers was 6 and the cord angle was 5°.

According to Table 2 and Fig. 5(c), when the number of cord layers was constant, the vertical stiffness of the membrane air spring increased with the enlargement of the cord cross-sectional area, exhibiting a positive correlation. When the number of cord layers was 2, 4, and 6, the increases in vertical stiffness as the cord cross-sectional area rose from 0.100 to 0.283 mm² were 19.917 N/mm, 30.976 N/mm, and 28.468 N/mm, corresponding to growth ratios of 63.3 %, 42.6 %, and 29.7 %, respectively. These results indicate that as the number of cord layers increased, the influence of enlarging the cord cross-sectional area on the growth ratio of vertical stiffness weakened gradually.

When the cord cross-sectional area was constant, vertical stiffness increased with the number of cord layers. For cross-sectional areas of 0.100, 0.166, and 0.283 mm², as the number of cord layers increased from 2 to 6, the vertical stiffness increased by 64.499 N/mm, 68.500 N/mm, and 73.050 N/mm, corresponding to increase ratios of 204.9 %, 164.0 %, and 142.1 %, respectively. This indicates that, as the number of cord layers increased from 2 to 6, a larger cord cross-sectional area enhanced the overall vertical stiffness but gradually reduced its effect on the growth rate of stiffness. The minimum vertical stiffness of 31.476 N/mm occurred when the number of cord layers was 2 and the cord cross-sectional area was 0.100 mm², while the maximum vertical stiffness of 124.443 N/mm occurred when the number of cord layers was 6 and the cross-sectional area was 0.283 mm².

5. Influence of cord parameter combination on the vertical stiffness of the membrane air spring

5.2. Orthogonal experimental range analysis

Range analysis, commonly represented by $R$, is a statistical method used to evaluate the relative influence of different factors. A larger $R$ value indicates a greater impact of the factor on the output variable. The average values of the nine experimental groups for each factor and level are denoted as $\stackrel{-}{K_{\dot{ij}}}$, where $i$ represents the level ($i=$ 1, 2, 3) and $j$ represents the factor ($j=t$, $m$, $\alpha$, $s$). The range $R$ is calculated as the difference between the maximum and minimum values of $\stackrel{-}{K_{\dot{ij}}}$ for each factor, as shown in Eq. (11):

11

$R=\mathrm{m}\mathrm{a}\mathrm{x}\left(\stackrel{-}{K_{\dot{ij}}}\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left(\stackrel{-}{K_{\dot{ij}}}\right).$

Using the vertical stiffness values from Table 4 and Eq. (11), the range analysis results were obtained and are presented in Table 5. Based on the calculated $\stackrel{-}{K_{\dot{ij}}}$ values, the influence trends of the four cord parameters at three levels were plotted, as shown in Fig. 6.

Table 4Calculation results of 4 factors 3 levels orthogonal experiments

Group number	$t$ (layer)	$m$ (mm)	$\alpha$ (°)	$s$ (mm2)	$K_1$ (N∙mm−1)
1	1	1	1	1	47.575
2	1	2	2	2	40.751
3	1	3	3	3	35.525
4	2	1	2	3	111.925
5	2	2	3	1	56.953
6	2	3	1	2	73.925
7	3	1	3	2	110.275
8	3	2	1	3	124.072
9	3	3	2	1	76.725

Table 5Range analysis table

Index	$t$ (layer)	$m$ (mm)	$\alpha$ (°)	$s$ (mm2)
$\stackrel{-}{K_1}$	41.283	89.925	81.857	60.418
$\stackrel{-}{K_2}$	80.934	73.925	76.467	74.983
$\stackrel{-}{K_3}$	103.691	62.058	67.584	90.507
R	62.408	27.867	14.273	30.089

Fig. 6Influence of four cord factors at three levels on vertical stiffness

As illustrated in Fig. 6, vertical stiffness increased significantly with the number of cord layers, indicating a strong positive correlation. As the cord cross-sectional area increased, vertical stiffness also increased, showing a relatively strong influence. Conversely, as cord spacing increased, vertical stiffness decreased, demonstrating a moderate negative effect. The influence of cord angle on vertical stiffness was the weakest, with only a minor decreasing trend observed.

According to Table 5 and Fig. 6, the order of factor influence based on the range $R$ was: $R_t>R_s>R_m>R_{\alpha }$. Using maximum vertical stiffness as the evaluation criterion, the maximum $\stackrel{-}{K_{\dot{ij}}}$ value in each factor was selected to combine the optimal level group $t_3{m}_1{\alpha }_1{s}_3$.

5.3. Comparative analysis of the optimal level combination

After reviewing relevant literature, the intermediate level combination $t_2{m}_2{\alpha }_2{s}_2$ was selected as a reference for comparison with the optimal level combination $t_3{m}_1{\alpha }_1{s}_3$, as shown in Table 3. The load-displacement curves of both configurations were obtained, as illustrated in Fig. 7.

Fig. 7Load-displacement curves of t3m1α1s3 and t2m2α2s2

As shown in Fig. 7, the overall load of both $t_3{m}_1{\alpha }_1{s}_3$ and $t_2{m}_2{\alpha }_2{s}_2$ combinations increased nonlinearly with displacement and the growth rate became progressively larger. At the same displacement, the load corresponding to $t_3{m}_1{\alpha }_1{s}_3$ was consistently higher than that of $t_2{m}_2{\alpha }_2{s}_2$. Furthermore, the difference in load between the two groups widened as the displacement increased. The maximum compression displacement for $t_3{m}_1{\alpha }_1{s}_3$ was 33.275 mm, with a corresponding load of 4169.438 N. A quartic polynomial fitting was applied to the optimal level group $t_3{m}_1{\alpha }_1{s}_3$ to obtain the functional relationship describing the variation law of vertical stiffness, as expressed in Eq. (12):

12

$f\left(x\right)={8.104x}^4-26.07x^3+166.3x^2+1244x+1420.$

6. Influence of membrane air spring cord parameters on 50 mm compression displacement

To further examine the variation characteristics of vertical stiffness under a finite and larger compression displacement, the compression displacement was increased to 50 mm. Based on the four cord parameters in Table 1 – namely, the number of cord layers, cord spacing, cord angle, and cord cross-sectional area – each parameter was divided into three groups, resulting in 81 experimental combinations. The corresponding vertical stiffness values were calculated for all groups, and the results are summarized in Table 6.

In Table 6, among the groups numbered 5, 6, and 9, the results of the four parameter combinations failed to converge. In these cases, the compression displacement z and the vertical stiffness $K_1$ are represented by $X$. There were 18 groups (numbers 19, 20, 22, 46, 47, 48, 49, 50, 64, 65, 66, 73, 74, 75, 76, 77, 78, and 79) that did not reach the full compression displacement of 50 mm.

From Table 6, when the number of cord layers $t$ was 2, 4, and 6, and the cord spacing $m$ and angle $\alpha$ were 3 mm and 5°, respectively, with a cord cross-sectional area s of 0.100 mm², the corresponding test groups were 1, 10, and 19. When s was 0.166 mm², groups 28, 37, and 46 were selected. When $s$ was 0.283 mm², groups 55, 64, and 73 were selected. The compression displacement $z$ and vertical stiffness $K_1$ for these nine groups were compared, as illustrated in Fig. 8.

As shown in Fig. 8, the horizontal axis represents the experimental group numbers, while the vertical axis corresponds to compression displacement $z$ and vertical stiffness $K_1$. When the cord spacing and angle were constant and the cord cross-sectional area values were 0.100, 0.166, and 0.283 mm², an increase in the number of cord layers led to a rise in vertical stiffness $K_1$, reaching its maximum at six layers. However, as the number of cord layers increased, the allowable compression displacement $z$ of the membrane air spring gradually decreased, indicating a trade-off between stiffness and deformability.

Fig. 8Influence of the number of cord layers on compression displacement z and vertical stiffness K1 of the membrane air spring

According to Table 6, when the cord spacing ($m$) was 3, 5, and 7 mm, and the number of cord layers $t$ and angle $\alpha$ were fixed at six layers and 5°, respectively, while the cord cross-sectional area $s$ was 0.100 mm², the selected groups were 19, 22, and 25. When $s$ was 0.166 mm², groups 46, 49, and 52 were selected, and when $s$ was 0.283 mm², groups 73, 76, and 79 were selected. The compression displacement $z$ and vertical stiffness $K_1$ for these nine groups were compared, as shown in Fig. 9.

As is illustrated in Fig. 9, the horizontal axis represents the group numbers, and the vertical axis indicates the compression displacement $z$ and vertical stiffness $K_1$, respectively. When the number of cord layers and the cord angle were constant, and the cord cross-sectional areas were 0.100, 0.166, and 0.283 mm², and the compression displacement $z$ that the membrane air spring could withstand decreased progressively with increasing cord spacing. Meanwhile, vertical stiffness $K_1$ also decreased as cord spacing increased, exhibiting a clear negative correlation between cord spacing and vertical stiffness.

Fig. 9Influence of cord spacing on compression displacement z and vertical stiffness K1 of the membrane air spring

According to Table 6, when the cord cross-sectional areas ($s$) were 0.100, 0.166, and 0.283 mm², and the number of cord layers ($t$) and the cord angle ($\alpha$) were fixed at six layers and 5°, respectively, with cord spacing ($m$) set to 3 mm, groups 19, 46, and 73 were selected. When $m$ was 5 mm, groups 22, 49, and 76 were used; when $m$ was 7 mm, groups 25, 52, and 79 were selected. The compression displacement $z$ and vertical stiffness $K_1$ for these nine groups were compared, as illustrated in Fig. 10.

Fig. 10Influence of the cord cross-sectional area on compression displacement and vertical stiffness of the membrane air spring

As shown in Fig. 10, the horizontal axis represents the experimental group number, while the vertical axis denotes the compression displacement $z$ and vertical stiffness $K_1$. When the number of cord layers and cord angle were constant, and the cord spacing values were 3, 5, and 7 mm, the compression displacement $z$ that the membrane air spring could withstand gradually decreased with an increase in the cord cross-sectional area. Meanwhile, vertical stiffness $K_1$ increased progressively with increasing cross-sectional area, demonstrating a positive correlation between the two.

According to Table 6, when the cord angle ($\alpha$) was 5°, 10°, and 15°, and the number of cord layers ($t$) and cord spacing ($m$) were fixed at six layers and 3 mm, respectively, with the cord cross-sectional area ($s$) set to 0.100 mm², groups 19, 20, and 21 were selected. When $s$ was 0.166 mm², groups 46, 47, and 48 were used, and when s was 0.283 mm², groups 73, 74, and 75 were taken. The compression displacement $z$ and vertical stiffness $K_1$ of these nine groups were compared, as shown in Fig. 11.

Fig. 11Influence of cord angle on compression displacement and vertical stiffness of the membrane air spring

As illustrated in Fig. 11, the horizontal axis represents the experimental group numbers, and the vertical axis represents compression displacement $z$ and vertical stiffness $K_1$. When the number of cord layers and cord spacing remained unchanged, and the cord cross-sectional areas were 0.100, 0.166, and 0.283 mm², the compression displacement $z$ that the membrane air spring could withstand increased gradually with the increase in cord angle. For groups 19, 20, and 21, vertical stiffness $K_1$ showed a decreasing trend with increasing cord angle, while for groups 46, 47, 48 and groups 73, 74, 75, vertical stiffness $K_1$ exhibited an increasing trend. These results indicate that the effect of cord angle on vertical stiffness is non-monotonic and may depend on the combination of other cord parameters.

Fig. 12Load-displacement curves of Group 8 and Group 80

Based on the data in Table 6, among all experimental groups that achieved a compression displacement of 50 mm, the combination of four cord parameters resulting in the smallest vertical stiffness corresponded to group 8, while the combination producing the largest vertical stiffness corresponded to group 80. The load-displacement curves for groups 8 and 80 were plotted and are presented in Fig. 12.

As shown in Fig. 12, the loads sustained by both groups increased with increasing displacement. Under the same compression displacement, the six-layer cord configuration withstood a higher load than the two-layer cord configuration. Furthermore, at each displacement level, the load value of group 80 was greater than that of group 8, and the difference in load between the two groups widened progressively as displacement increased.

7. Conclusions

To investigate the vertical stiffness characteristics of membrane air springs within a confined installation space, the influence of rubber airbag cord parameters on vertical stiffness was analyzed. A three-dimensional model of the membrane air spring was developed using SolidWorks, and its vertical stiffness behavior was simulated through ABAQUS. The effects of four key cord parameters – number of cord layers, cord spacing, cord angle, and cord cross-sectional area – on the vertical stiffness were systematically examined. The main conclusions are summarized as follows:

1) The influence characteristics of the four cord parameters on the vertical stiffness of membrane air springs were analyzed individually. When three parameters remained constant and only one was varied, the vertical stiffness of the membrane air spring decreased with increasing cord spacing and cord angle, while it increased with larger cord cross-sectional area and greater number of cord layers.

2) Based on the orthogonal experimental design method, the relative importance of the four cord parameters influencing vertical stiffness was determined. The number of cord layers exhibited the most significant effect on vertical stiffness, with an increase in layer count producing a pronounced enhancement in stiffness. The effects of cord spacing and cord cross-sectional area were moderate in comparison, while the cord angle had the least influence.

3) When the compression displacement was set to 50 mm, the combined influence of the four cord parameters on vertical stiffness was analyzed, and the parameter combinations yielding the minimum and maximum stiffness values were identified. The minimum vertical stiffness occurred when the number of cord layers was 2, the cord spacing was 7 mm, the cord angle was 10°, and the cord cross-sectional area was 0.100 mm². The maximum vertical stiffness was obtained when the number of cord layers was 6, the cord spacing was 3 mm, the cord angle was 15°, and the cord cross-sectional area was 0.283 mm².

The above research findings offer theoretical support for subsequent bench testing. Moving forward, bench tests targeting the membrane air spring will be carried out systematically. Drawing on the ensuing test data and analytical, it will provide references for the structural optimization and performance enhancement of membrane air spring.