Design of Fiber Winding for Thick-Walled Cylinder with Uniform Residual Tension

Abstract

Fiber winding with metal cylinders as liners is the primary method used for producing high-speed rotors, flywheels, and pressure vessels. A design model was established for thick-walled cylindrical fiber winding by using the layer-by-layer stacking method of thin composite material rings in this article. The effect of liner structure and fiber type on prestress was analyzed. And the phenomenon of tension loss in no-twist fibers was studied. The winding tension of each layer was calculated accurately under the condition of uniform residual tension, which was the optimal process in engineering. Compared with the experimental results, the computational error of the model does not exceed 5%. The winding model developed in this article, which takes into account tension loss, can more accurately guide the design of winding tensions for thick-walled cylinders.

1 Introduction

The main technique in the formation of high-speed rotors, energy storage flywheels, pressure vessels, and other mechanical equipment is fiber tension winding with metal cylinders as liners. The longevity and dependability of the spinning or bearing elements are directly impacted by whether the tension distribution of each layer and the winding prestress fulfill the design criteria. In recent years, with the development of new pressure vessels that are resistant to high pressure, corrosion and irradiation, the wall thickness of the lining has been designed to be thicker and thicker, and fibre-wound reinforcement technology based on thick-walled cylinders has become an industry requirement.

Since the 1990s, several academics have studied fiber winding in great detail. For the solution of residual tension, Springer and Calius [1, 2] suggested a multi-layer composite thin-walled cylinder overlaid model. A more precise fiber winding model that takes the resin flow process into account was created by Gutowski [3]. Cohen [4, 5] examined how factors including resin, fiber type, winding tension distribution, and winding angles affected the structure's stress. Zhao [6] performed the FEA modeling of fiber wet winding and examined several factors throughout the winding and curing process, including temperature, degree of curing, resin viscosity, and others. Based on the thin-shell model, Wang [7, 8] derived the tension formula for fiber winding with a thin-walled cylinder. The impact of matrix penetration, threshold tension, and other variables on the precision of prestress calculations for thin-walled cylinders was determined by Lu [9]. As a result of his investigation into the connection between interference amount and prestress, Qin [10] arrived at the composite flywheel winding's simplified prestress model. Ren [11, 12] suggested an intricate and very precise iterative approach for calculating the fiber tension of each layer under the assumption of uniform residual stress. Based on the idea of elastic deformation of an anisotropic winding layer and thick-walled cylinder with isotropic liner, Kang [13] suggested an analytical technique of residual tension. In a research on the dynamic properties of composite fiber tension wound prestress fields, Su [14] provided an explanation of the stress saturation phenomena.

In general, a lot of studies has been done on fiber winding in the theoretical, modeling, and experimental realms, but there are still two issues that have not received thorough investigation. First, thin-walled cylinders are the subject of the majority of investigations, and the thin-shell model is used to determine their winding tension. There are few studies on thick-walled cylinders. For thick-walled cylinders, the designing technique for winding tension often relies on intricate, layer-by-layer iterative calculations or repetitive process testing. It lacks a precise formula that can be applied to direct calculations. Second, while designing winding tension, the tension loss issue is typically disregarded. It is necessary to increase the precision of prestress design, particularly for non-twist fibers like carbon fiber whose tension loss is more pronounced than that of twist fibers like glass fiber. A technique was provided in the study to determine the starting tension of each layer for a thick-walled cylinder under the assumptions of uniform residual tension in order to address the aforementioned issues. A series of process tests led to the establishment of the winding tension loss model and a modification of the tension designing formula, which served as guidelines for designing the winding tension of thick-walled cylinders.

2 Fiber Winding Theory

2.1 Winding Model

After each layer of winding in the fiber winding process, the surface pressure of the liner and the stress distribution of the wound fibers alter. Figure 1 depicts the fiber winding geometric model.

Fig. 1

Fiber winding geometric model

Elastic modulus E_m, Poisson's ratio ν_m, inner radius r_mi, and outer radius r_mo are the parameters of a metal liner. Inner radius r_fi, outer radius r_fo, radial and circumferential modulus E_θ and E_r, Poisson's ratio μ_θr and μ_rθ are the parameters of the winding layer.

The thickness Δh of the fiber layer is wound under tension T(h), resulting in new displacement and stress to fulfill the equation.

$$\frac{{d\Delta \sigma_{r} }}{dr} + \frac{{\Delta \sigma_{r} - \Delta \sigma_{\theta } }}{r} = 0$$

(1)

Orthogonal anisotropic composite ring's geometric and physical equations are,

$$\begin{array}{*{20}c} {\Delta \varepsilon_{r} \left( r \right) = d\Delta u/dr} \\ {\Delta \varepsilon_{\theta } \left( r \right) = \Delta u/r} \\ {\Delta \varepsilon_{r} = \Delta \sigma_{r} /E_{r} - \mu_{\theta r} \Delta \sigma_{\theta } /E_{\theta } } \\ {\Delta \varepsilon_{\theta } = \Delta \sigma_{\theta } /E_{\theta } - \mu_{r\theta } \Delta \sigma_{r} /E_{r} } \\ \end{array}$$

(2)

The border conditions are,

$$\begin{array}{*{20}c} {\Delta \sigma_{r} \left( h \right) = - T\left( h \right)\Delta h/h} \\ {\Delta \sigma_{r} \left( {r_{i} } \right) = \Delta \sigma_{rm} \left( {r_{mo} } \right)} \\ \end{array}$$

(3)

The variation of radial displacement, stress, and surface pressure of each wound layer may be solved using Eq. (1). By integrating, one may determine the total interface pressure (P), radial displacement (u), and stress (σ).

$$\begin{array}{*{20}c} {P\left( h \right) = 2\lambda r_{mo}^{\lambda - 1} f\left( {r_{mo} } \right)} \\ {u\left( r \right) = \left[ {\left( {\lambda - \mu_{\theta r} - K} \right)\left( {\lambda + \mu_{\theta r} } \right)\left( {\frac{{r_{mo} }}{r}} \right)^{\lambda } - \left( {\lambda + \mu_{\theta r} + K} \right)\left( {\lambda - \mu_{\theta r} } \right)\left( {\frac{r}{{r_{mo} }}} \right)^{\lambda } } \right]\frac{{r_{mo}^{\lambda } }}{{E_{\theta } }}f\left( r \right)} \\ {\sigma_{r} \left( r \right) = - \left[ {\left( {\lambda - \mu_{\theta r} - K} \right)\left( {\frac{{r_{mo} }}{r}} \right)^{\lambda + 1} + \left( {\lambda + \mu_{\theta r} + K} \right)\left( {\frac{r}{{r_{mo} }}} \right)^{\lambda - 1} } \right]r_{mo}^{\lambda - 1} f\left( r \right)} \\ {\sigma_{\theta } \left( r \right) = \left[ {\left( {\lambda - \mu_{\theta r} - K} \right)\left( {\frac{{r_{mo} }}{r}} \right)^{\lambda + 1} - \left( {\lambda + \mu_{\theta r} + K} \right)\left( {\frac{r}{{r_{mo} }}} \right)^{\lambda - 1} } \right]\lambda r_{mo}^{\lambda - 1} f\left( r \right) + T\left( r \right)} \\ \end{array}$$

(4)

In the equation above, the integral function's expression is,

$$f\left( r \right) = \int {h^{\lambda } T\left( h \right)/\left[ {\left( {\lambda + \mu_{\theta r} + K} \right)h^{2\lambda } + \left( {\lambda - \mu_{\theta r} - K} \right)r_{mo}^{2\lambda } } \right]} dh$$

(5)

Each coefficient's expression is,

$$\begin{array}{*{20}c} {K = E_{\theta } \left[ {\left( {1 - \upsilon_{m} } \right) + \varsigma^{2} \left( {1 + \upsilon_{m} } \right)} \right]}/E_{m}(1-\varsigma^{2}) \\ {\varsigma = r_{mi} /r_{mo} } \\ {\lambda = \sqrt {E_{\theta } /E_{r} } } \\ \end{array}$$

(6)

From the surface pressure P(h), it is simple to compute the displacement and stress of an isotropic mental liner [15]. Wet winding is currently a widely utilized production process. The equivalent stiffness of the uncured wound layer may be determined using the empirical formula provided by Cai and Gutowski.

$$\begin{array}{*{20}c} {E_{\theta } = \pi E_{f} V_{f} /4V_{a} } \\ {E_{r} \frac{{A_{s} \sqrt {V_{a} /V_{0} } \left[ {5 - \sqrt {V_{f} /V_{a} } - 4\sqrt {V_{0} /V_{f} } } \right]}}{{2\left( {\sqrt {V_{a} /V_{f} } - 1} \right)^{5} }}} \\ \end{array}$$

(7)

Taking into account the variation in fiber volume content with radius, λ in formula (6) should be recast as λ(h), which is more precise as an integral function, but the computation is also trickier.

2.2 Design with Uniform Residual Tension

The inner fiber relaxes once the outer fiber has been wrapped. Making the residual tension of each layer after winding equal is the technological goal sought in practical engineering. N represents the overall number of winding layers. The illness is defined as,

$$\sigma \left( {r_{i} } \right) = \sigma \left( {r_{i - 1} } \right)\quad \left( {i = 2,3, \ldots ,N} \right)$$

(8)

By combining (4) and (8), equations relating to the winding tension may be rewritten as,

$$MT = H$$

(9)

T represents each layer's winding tension.

$$T = \left[ {\begin{array}{*{20}c} {T_{1} } & {T_{2} } & \ldots & {T_{i} } & \ldots & {T_{N} } \\ \end{array} } \right]^{T}$$

(10)

M is the coefficient matrix.

$$\left[ {\begin{array}{*{20}c} {B_{1} } & {B_{2} } & {B_{3} } & \ldots & {B_{i} } & \ldots & {B_{N} } \\ { - A_{1} B_{1} + 1} & {\left( {A_{2} - A_{1} } \right)B_{2} r_{2}^{\lambda } - 1} & {\left( {A_{2} - A_{1} } \right)B_{3} } & \ldots & {\left( {A_{2} - A_{1} } \right)B_{i} } & \ldots & {\left( {A_{2} - A_{1} } \right)B_{N} } \\ {} & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ {} & { - A_{j} B_{j} + 1} & {\left( {A_{j} - A_{j - 1} } \right)B_{j} r_{j}^{\lambda } - 1} & \ldots & {\left( {A_{j} - A_{j - 1} } \right)B_{i} } & \ldots & {\left( {A_{j} - A_{j - 1} } \right)B_{N} } \\ {} & {} & \ldots & \ldots & \ldots & \ldots & \ldots \\ {} & {} & {} & {} & { - A_{N - 2} B_{N - 2} + 1} & {\left( {A_{N - 1} - A_{N - 2} } \right)B_{N - 1} r_{N - 1}^{\lambda } - 1} & {\left( {A_{N - 1} - A_{N - 2} } \right)B_{N} } \\ 0 & {} & {} & {} & {} & { - A_{N - 1} B_{N - 1} + 1} & {\left( {A_{N} - A_{N - 1} } \right)B_{N} r_{N}^{\lambda } - 1} \\ \end{array} } \right]$$

(11)

The expression of matrix H is,

$$H = \left[ {\begin{array}{*{20}c} {\frac{{1 - r_{mi}^{2} /r_{mo}^{2} }}{{1 + r_{mi}^{2} /r_{mo}^{2} }}\frac{{\sigma_{0} }}{{2\lambda r_{mo}^{\lambda - 1} }}} \\ 0 \\ \cdots \\ 0 \\ \end{array} } \right]$$

(12)

The definition of Coefficient A and B is,

$$\begin{array}{*{20}c} {A_{i} = \left[ {\left( {\lambda - \mu_{\theta r} - K} \right)\left( {\frac{{r_{mo} }}{{r_{i} }}} \right)^{\lambda + 1} - \left( {\lambda + \mu_{\theta r} + K} \right)\left( {\frac{{r_{i} }}{{r_{mo} }}} \right)^{\lambda - 1} } \right]\lambda r_{mo}^{\lambda - 1} } \\ {B_{i} = \frac{{r_{i} - r_{i - 1} }}{{\left( {\lambda + \mu_{\theta r} + K} \right)r_{i}^{2\lambda } + \left( {\lambda - \mu_{\theta r} - K} \right)r_{mo}^{2\lambda } }}r_{i}^{\lambda } } \\ \end{array}$$

(13)

A fiber tension algorithm that satisfies the requirement of uniform residual tension was developed. The real tension of the fiber is lower than the technical tension during the winding process of no-twist fiber, such as carbon fiber. The utilization coefficient of fiber tension is defined as,

$$\delta_{i} = \frac{{T_{actl} }}{{T_{tech} }} = \sqrt {\frac{{1 + \left( {i - 1} \right)\chi }}{1 + n\chi }}$$

(14)

The equivalent thickness of the resin matrix χ is an empirical value that is established by tests. A more precise algorithm for designing tension has been developed by transforming the technological tension (T_tech) into the actual tension (T_actl).

3 Numerical Examples

3.1 Constant Technological Tension Winding

The distribution law of the prestress in a thick-walled cylinder was examined using constant technical tension winding as an example. Table 1 displays the winding process's parameters.

Table 1 Winding process’s parameters

Figure 2 depicts the stress distribution under various winding tensions. It is demonstrated that whereas the circumferential stress jumps at the interface between the liner and the wound layer, the radial stress fluctuates constantly throughout the radial direction. With a rise in winding tension, the liner's prestress grows quickly.

Fig. 2

Stress distribution of liner and wound layers

Figure 3 depicts the relationship between the prestress, the thickness of the liner, the number of wrapped layers, and the fiber modulus. It is seen in Fig. 3:

Fig. 3

Relationship between prestress and winding parameters

• (1) The bigger the prestress, the thinner the liner and the higher the winding tension. In this instance, the wound prestress is decreased by nearly half while the liner thickness is raised from 1 to 2 mm. The thick-walled cylinder's prestressing is quite challenging.

• (2) Stress saturation is the situation where the impact of prestressing the liner by winding steadily decreases with the number of wrapped layers. It is more important for the method when the wound layer's circumferential modulus is higher and the liner modulus is lower. It is best to wind fiber with a low modulus to increase prestress.

Combining the process circumstances, liner material, liner thickness, working load, structural strength, and failure mode will help you choose the right fiber and tension.

3.2 Variable Volume Content of Fiber

Take the linear distribution of fiber volume content along the radial direction as an example. Figure 4 depicts the impact of the change in volume content on the stress distribution.

Fig. 4

Influence of fiber volume content on stress

We may infer the following from Fig. 4:

• (1) As the gradient of the fiber volume content increases, the liner's prestress reaches an extreme point. The prestress's adjustable range, however, is constrained. Therefore, the basic engineering assumption of homogeneous volume content can still be accurate enough.

• (2) The radial stress of the wound layer remains essentially constant as the fiber volume content gradient grows, whereas the degree of nonlinearity of the circumferential stress increases. It should be taken into account when winding stress is precisely designed.

3.3 Uniform Residual Tension

The design scheme for the thick-walled cylinder example from the literature [11] is illustrated in Table 2.

Table 2 Comparison of winding tension results

The results in this article are in good agreement with the literature, and the highest variance is less than 1%, demonstrating the high precision of the algorithm used here, which is also easier to use than the iterative approach used in the literature.

4 Experimental Verification of the Tension Loss Model

Three groups of winding experiments with various liner thicknesses and fiber types were created in order to validate the accuracy of the tension loss model put out in the research. Table 3 compares the experimentation circumstances in each group.

Table 3 Comparison of experiment parameters

The dynamic strain measurement apparatus was utilized to assess how prestress changed layer by layer while the fiber was wound (Fig. 5). The inner wall of the liner's liner was evenly covered with four strain gauges, A, B, C, and D. The mean value of the strain was taken as the liner's circumferential strain.

Fig. 5

Dynamic strain monitoring device

The circumferential strain of the liner layer by layer during fiber winding was calculated using the tension loss model described in the study, and the comparison with the test results is shown in Fig. 6.

Fig. 6

Test results of strain

Figure 6 demonstrates that, with practically no tension loss, the real tension of glass fiber winding is nearly equivalent to the technical tension. However, the carbon fiber winding tension loss is rather evident. There is a significant discrepancy between the real prestress and the design if the winding tension is estimated while disregarding tension loss. The largest strain divergence between the modified designs and the testing in #2 and #3, according to the tension loss model, was 4.1% and 3.3%, respectively. The experiments and the theoretical prediction correspond rather well.

5 Conclusions

Winding tension was designed for thick-walled cylinders with uniform residual tension in this thesis. The following are the primary conclusions:

• (1) A theoretical model of fibre tension winding in thick-walled cylinders is established, the analytical formulae for fibre winding tension and liner prestress distribution are obtained, and a direct design method for the tension of each layer of equal tension winding is given. The calculated results do not deviate by more than 1% compared with the design in the relevant literature.

• (2) The thinner the liner and the lower the fibre modulus, the better the prestressing effect. The design of the number of wrapped layers needs to take into account the stress saturation phenomenon. The difference in the volume content of the inner and outer fibers does not have a significant effect on the prestress.

• (3) There is basically no loss in the winding tension of glass fibers, while the actual tension of carbon fibers is significantly lost compared to the process tension. The winding tension loss model proposed in this paper can predict the actual prestress more accurately, and the deviation is within 5% compared to the experimental results.