Research on molding method and thermal-dependent tension system based on heated-mandrel winding process

Abstract

Heated-mandrel winding is a typical application of in situ curing process which realizes the integration of winding and curing process, which has been widely used in the molding process of composite materials. However, the new problems have been raised regarding the application of this process in engineering practice. There is a strong coupling relationship between the temperature and curing degree, and the coupling results will affect the distribution of stress field in the molding process. The original residual stress balance between the adjacent layers will be destroyed and the delamination, wrinkle, or fracture will occur in the molded shell. To solve this problem, this paper considers the impact of temperature-curing field changes on the thermal parameters during winding and the process of winding with tensioning is equivalent to the interference fit of thin layers. A linearized tension calculation model is established after summarizing the variation law of pre-stress with coupling field. The influence of the coupling field on stress balance is compensated by controlling and adjusting the winding tension to ensure that the radial deformation of each winding layer under pressure load is controllable. The results show that the numerical simulation results are in a good agreement with the experimental data.

Introduction

As a new type of pollution-free, green, and new energy technology, flywheel is widely used in the field of energy storage because of its high energy storage efficiency, high energy storage density, strong adaptability, economic, and environmental protection. ¹ In the early days, flywheels were mostly made of metal materials, which were suitable for the applications with low-speed requirements. However, with the maturity of flywheel energy storage technology, its application field is expanding. It is necessary to increase the speed of the flywheel to maximize the energy storage. Since the speed of the energy storage flywheel is rather high, it is very important to reduce its own weight to improve the speed. Compared with the metal flywheel, composite flywheel has lighter weight, higher structural strength, and better performance of preventing fracture surface expansion. Therefore, the composite flywheel is more suitable for the requirements of new fields. The emergence of composite flywheel makes up for the shortcomings of these aspects. It shows the advantages of high specific strength, high specific stiffness, light weight, and long life in stiffness and strength.

In the winding process, the radial strength has become a key factor restricting the flywheel to achieve higher speed. Therefore, the control of winding tension in the winding process is an effective way to improve the radial strength for the flywheel. ² Cohen ³ studied the effect of winding tension on the mechanical properties of composite shell by using experimental methods. He found that an appropriate increase in winding tension is beneficial to increase the volume fraction of shell to improve the bearing capacity of the molded product. Zobeiry ⁴ established a residual stress pseudo-viscous elastic model to improve the calculation efficiency in the curing stage for a thermosetting composite. Ding ⁵ derived a metal cylinder winding tension formula based on the de-flection theory for a thin-walled cylinder, considering the relaxation inside the wound fiber layers caused by the layers that were being wound. Ren ⁶ used the method of combination of the virtual and real, and iterative search to solve the equal tension winding problem. The stress field caused by winding tension is simulated and numerically solved by the layer-by-layer finite element method to ensure that the pre-tension value of each fiber layer after winding is the same. Zhang ⁷ developed a variable tension system with decreasing tension gradient to improve the tensile properties of NOL ring. Wei ⁸ took the pressure vessel as the research object, developed the calculation formula of the winding tension of the innermost layer and the other winding layers by using the method of spiral combined hoop winding, and studied the relationship between the optimal pre-stress and the winding tension of the filament wound composite pressure vessel under the elastic and semi-elastic design conditions. Therefore, the commonly used method in many mechanical studies on winding process is to establish a reasonable tension system to form an equal amount of radial residual stress between the winding layers inside the molded flywheel, improve the compressive and deformation resistance of the molded products, and enhance the overall load-bearing capacity of the flywheel. For this reason, Xing ⁹ studied the hoop winding tension on a rigid cylinder. The relaxation relationship between each winding layer was considered based on the constitutive equation of isotropic material, an analytical method for solving the tension distribution of hoop winding on a rigid cylinder was derived, and a gradient tension system design method based on equal residual winding tension was given. Liu ¹⁰ proposed a residual tension analysis method and winding tension design method considering the influence of mandrel deformation. According to the three-dimensional constitutive relationship of isotropic materials and the thick-walled cylinder theory of elastic mechanics, the stress and strain of the winding layer under external pressure are obtained by analytical method. Ren ¹¹ proposed a design method based on the winding process where the tension value imposed on any winding layer can be calculated only to define the initial winding tension of that winding layer. Kang ¹² used the superposition principle to establish an analytical algorithm between the residual tension and the winding tension within the elastic range and gave the calculation model of radial stress and hoop stress of the winding layer under external pressure based on the elastic deformation of the anisotropic winding layer and the theory of thick-walled barrels. Calius ¹³ established an analytical calculation formula and used it to simplify the complex mechanical model. Cohen ¹⁴ identified the winding tension as one of the most significant production parameters affecting the mechanical performance of the molding shell using some experiments, and he concluded that it was vitally important for the compression capacity of the molding shell to control the winding tension. However, most of these existing studies are based on the traditional single winding process, which itself has problems such as the product volume is limited and the stress balance between adjacent layers is difficult to control in the later stage of the heating and curing process. These problems can be solved in the heated-mandrel winding process. In the heated-mandrel winding process, the wound composite shell does not need to be moved, and the in situ curing can be realized which can solve the problem of product volume limitation. In addition, the imbalance of residual stress between adjacent layers caused by the curing reaction in the winding stage can be compensated by controlling the winding tension in the heated-mandrel winding process. Therefore, the tension system based on the single winding process in which the variations of parameter state caused by curing reaction can be neglected in the winding stage since the winding and curing are separated is difficult to apply to the heated-mandrel winding process since the two molding processes are rather different.

This paper proposes a tension analysis method which is suitable for the heated-mandrel winding process and establishes a tension system model based on the heated-mandrel winding process. In the process of solving the tension system model, the strong coupling relationship between the temperature field and the curing field is qualitatively analyzed, and the influence of thermal parameters (such as the curing degree, the elastic modulus of resin, and the thermal expansion coefficient of composite) on the residual stress between adjacent layers is analyzed. The relationship between the winding pre-stress and winding tension is quantitatively calculated to realize the equal radial residual stress between adjacent winding layers after molding. At the same time, the nonlinear tension solution problem is transformed into a linear displacement–stress calculation problem by using the method of equivalent the winding with applying tension process to the interference fit of thin layers to avoid the complex nonlinear mathematical iterative search calculation, which theoretically simplifies the lengthy and complex traditional tension calculation.

Heated-mandrel winding process principle

In the heated-mandrel winding process, the heat can be transferred to the winding fiber layer by heating the metal mandrel with hot steam to realize the winding process accompanied by heating and curing, and the molding efficiency of the composite is accelerated. During the continuous winding with heating, the internal stress relationship of each winding layer can be effectively adjusted by controlling the winding tension system and heating curing system to maintain the stress balance between each winding layer. The winding–curing molding principle of heated-mandrel winding process is shown in Figure 1.

Figure 1.

Winding–curing molding principle of heated-mandrel winding process.

In the single winding stage of the traditional two-step molding process, the pre-stress of each winding layer is the radial component of the winding tension (i.e., the radial external pressure stress), which can be obtained by the equivalent calculation of the radial interference of the adjacent winding layers under the winding tension. To maintain the stress balance relationship between adjacent winding layers, the method of compensating pre-stress can be used to ensure that the adjacent layers are in a state of equal radial residual stress, and then the tension system can be derived based on the relationship between pre-stress and winding tension. In fact, the heated-mandrel winding process only simplifies the operation steps of the two-step molding process, while the tension control principle in the forming process remains basically unchanged. It is still based on the premise of equal residual stress between the adjacent layers to ensure that each winding layer is in a balanced state of stress after winding. In this way, the molded flywheel can become a uniformly stressed entity. Therefore, in the heated-mandrel winding process, the equal radial residual stress of adjacent winding layers is taken as a key point of the research, and the preset radial stress is quantitatively calculated to compensate the external pressure stress of winding to realize the control and adjustment of winding tension and obtain the optimal tension system. Meanwhile, when the winding process is accompanied by a curing reaction, it is necessary to consider the influence of the curing system on the molding process, that is, the equilibrium relationship of the original radial residual stress will be affected by the temperature-curing degree coupling field. Therefore, in the molding process, it is necessary to clarify the relationship between the radial residual stress and factors such as the temperature, curing degree, and radial external pressure stress for qualitative analysis and finally establish a tension system model suitable for heated-mandrel winding process. The stress analysis between adjacent layers in the heated-mandrel winding process is shown in Figure 2.

Figure 2.

Stress analysis between adjacent layers in the heated-mandrel winding process.

Analysis of curing process

In the heated-mandrel winding process, the winding process will be accompanied by the curing reaction, so there is a strong coupling relationship between the temperature field and the curing degree field in the curing process. That is, the temperature will affect the curing reaction effect, and the heat release of the curing reaction will also affect the temperature, so the relationship is rather complex. The relationship between temperature and curing degree can be described by curing system, and the curing system is determined by curing process. So, the curing process has a great influence on the quality of molding. In order to develop a reasonable curing process, this paper analyzes the curing system that affects the curing process of composite materials, explores the influence of temperature field changes on the thermal parameters in the curing stage, and provides a theoretical basis for the reasonable setting of curing parameters in the heated-mandrel winding process. A mathematical model that can reflect the changes of residual stress in the winding process is established, which lays a theoretical foundation for the formulation of thermal-dependent tension system suitable for heated-mandrel winding process.

Curing system

In heated-mandrel winding process, it is necessary to heat the composite materials according to a certain curing system to ensure the molding quality. Therefore, the relationship between heating temperature and heating time can be determined according to the requirements. That is, every certain time point will correspond to a certain temperature point, and this temperature point will correspond to a certain value of curing degree.15 The curing system curve is shown in Figure 3.

Figure 3.

Curing system curve.

Thermal parameters

Curing degree of resin matrix

In the heated-mandrel winding process, the curing reaction will occur when the resin matrix is heated to the curing temperature, and a large amount of heat will be released, which further increases the curing temperature. As the temperature gets higher, the curing reaction becomes more intense until all the resin matrix components complete the reaction. That is, the curing degree of the resin matrix ${\alpha }_{\mathrm{cure}}$ will change with time, as shown in Figure 4.

${\alpha }_{\mathrm{cure}}=1-{\left[A_0\hspace{0.17em}\exp \left(-\frac{E_a}{RT}\right)\left(n-1\right)t+1\right]}^{\frac{1}{1-n}}$

(1)

where $E_a$ is the activation energy, $A_0$ is the pre-exponential factor, and $n$ is the reaction progression.16,17

Figure 4.

Relationship between curing degree and curing time.

Different test points are selected at different positions to obtain the more accurately simulation results. The nodes 1, 2, 3, and 4 are selected from inner to outer of the model in turn for temperature measurement. The fiber grating sensor is used to measure the temperature of each selected test point in real time. ¹⁸ It communicates with PC through Ethernet with a 16-channel expansion module. Enlight (V1.5) signal analysis software was used for data acquisition and analysis, as shown in Figures 5 and 6. ¹⁵

Figure 5.

Bragg grating sensor implantation method.

Figure 6.

Optical sensing interrogator SM130.

Four fiber grating sensors are embedded in four different positions of the composite flywheel. At the same time, four grating temperature sensors (encapsulated in capillary tube) are embedded near the fiber grating sensors to compensate the wavelength variation caused by temperature. Two temperature sensors are pasted on the mandrel to monitor the temperature. The collected data are exported to obtain the wavelength data at different times. The wavelength data are used to minus the zero wavelength (the initial wavelength of the fiber grating sensor when it is not subjected to strain load) and the wavelength caused by temperature variation. Finally, the strain values caused by stress at different times can be obtained. The obtained strain values are substituted into the strain characteristic equation (2). The temperature sampling points and temperature–time curve of each monitoring point can be obtained, as shown in Figure 7.

$\mathrm{\epsilon }=\mathrm{\gamma }·∆\mathrm{T}$

(2)

Figure 7.

Temperature sampling points and temperature–time curve.

Where $\mathrm{\gamma }$ represents the thermal expansion coefficient.

Elastic modulus of composite materials

In the curing reaction, the state of the resin matrix will transition from a low modulus and uncured viscous flow state in the warm-up stage to a mixed viscous flow-viscoelastic state in the first half of the heating stage and a viscoelastic state in the second half of the heating stage, and finally a high modulus and fully cured glassy state flywheel is molded in the cooling stage, as shown in Table 1.

Table 1.

Change process of resin matrix state. ¹⁹

Curing stage	Control mode	Control time (/s)	Control temperature (°C)	Resin matrix state	Curing degree (%)
Warm-up stage	Heating	0–260	30–130	Viscous flow	30
	Warming	260–580	130
Heating stage	Heating	580–790	130–205	Viscous flow-viscoelastic	60
	Warming	790–1500	205
	Heating	1500–1750	205–300	Viscoelastic	90
	Warming	1750–2260	300
Cooling stage	Cooling	2260–2800	300–30	Glassy	100

Therefore, the elastic modulus of the composite materials will also change with temperature in the molding process and ultimately reach a stable state, ²⁰ as shown in Figure 8.

Figure 8.

Relationship between elastic modulus and curing time.

According to the basic mixing rule, the elastic modulus of the composite can be depicted as follows:

$E_c=E_f{V}_f+E_{\mathrm{h}}\left(1-V_f\right)$

(3)

where $E_f$ is the elastic modulus of the fiber, $V_f$ is the volume fraction of the fiber, and $E_h$ is the elastic modulus of the resin.

The elastic modulus of the resin $E_h$ varies constantly with the curing reaction during curing. Therefore, the elastic modulus $E_h$ related to the curing process can be expressed by the mixing rules. ²¹

$E_h=\left(1-{\alpha }_{\mathit{mod}}\right)E_h^0+{\alpha }_{\mathit{mod}}{E}_h^{\infty }$

(4)

${\alpha }_{\mathit{mod}}=\frac{\left(\alpha -{\alpha }_{\mathrm{gel}}\right)}{\left({\alpha }_{\mathrm{end}}-{\alpha }_{\mathrm{gel}}\right)}$

(5)

where ${\alpha }_{\mathrm{gel}}$ is the curing degree of the resin at the gel time, ${\alpha }_{\mathrm{end}}$ is the curing degree of the resin at the cured time, $E_h^0$ is the elastic modulus of the resin at the uncured time, $E_h^{\infty }$ is the elastic modulus of the resin at the cured time, and $E_M$ is the elastic modulus of the metal mandrel. The elastic modulus corresponding to any curing degree of the resin can be calculated during curing by the above equations. ¹⁵

$E_h=\left[\frac{E_h^0}{E_M}+{\alpha }_{\mathit{mod}}\left(\frac{E_h^{\infty }}{E_M}-\frac{E_h^0}{E_M}\right)\right]E_M$

(6)

The elastic modulus of the composite that changes with the curing degree can be obtained by substituting equation (6) into equation (3).

$E_c=E_f{V}_f+\left[\frac{E_h^0}{E_M}+{\alpha }_{\mathit{mod}}\left(\frac{E_h^{\infty }}{E_M}-\frac{E_h^0}{E_M}\right)\right]\left(1-V_f\right)E_M$

(7)

Mixed thermal expansion coefficient of composite

Since the mismatch of the thermal expansion coefficient of the fiber and the resin matrix, the corresponding residual stress will be generated in the matrix near the interface area of the fiber and resin matrix in variable temperature state, that is, the residual stress existing in the adjacent winding layers mentioned in the paper, which will destroy the original stress balance between the winding layers. Therefore, the calculation of the mixed thermal expansion coefficient $\mathrm{\gamma }$ of the composite materials is very important for the reasonable formulation of the tension system in the heated-mandrel winding process. The calculation formula is shown as follows: ²²

$\mathrm{\gamma }=\left({\epsilon }_f+v_f{\epsilon }_f\right)V_f+\left({\epsilon }_h+v_h{\epsilon }_h\right)\left(1-V_f\right)-\left[v_f{V}_f+v_h\left(1-{\hspace{0.17em}V}_f\right)\right]\frac{{\epsilon }_f{E}_f{V}_f+{\epsilon }_h{E}_h\left(1-V_f\right)}{E_f{V}_f+E_h\left(1-V_f\right)}$

(8)

where the subscript $f$ represents the material properties of the fiber, the subscript $h$ represents the material properties of the resin matrix, $\epsilon$ represents the curing shrinkage strain, and $v$ represents Poisson‘s ratio.

It is easy to cause the plastic deformation in adjacent winding layers by thermal stress in the heating stage of composite molding process. With the increase of time and temperature, the elastic modulus $E_h$ and the thermal expansion coefficient $\mathrm{\gamma }$ will also increase. ²³ The effect of changes of elastic modulus on $\mathrm{\gamma }$ is shown in Figure 9.

Figure 9.

Effect of the change of elastic modulus on $\mathrm{\gamma }$ .

Curing model

The changes of temperature field will cause the curing reaction of resin, which will affect the curing degree and curing reaction rate of composite materials in the heated-mandrel process. At the same time, the heat released by the curing reaction as an internal heat source will further affect the distribution of temperature field, so there is a strong coupling relationship between temperature and curing degree. The curing model can reflect the changes of temperature and curing degree of composite materials in real-time and provide a theoretical analysis basis for solving the thermal-dependent winding tension field.

Thermal conduction model

The temperature field of the curing process of composite flywheel is determined by the heat transfer and the heat release of the curing reaction. According to the thermal conduction mechanism and the energy conservation equation, the thermal conduction model is established as follows: ²⁴

$\rho c\frac{\partial T}{\partial t}=\frac{\partial }{\partial x}\left(k_x\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k_y\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left(k_z\frac{\partial T}{\partial z}\right)+H_r\rho \frac{d{\alpha }_{\mathrm{cure}}}{dt}$

(9)

where $\rho$ is the density of the composite, $c$ is the specific heat of the material, $T$ is the absolute temperature of the composite, $k_x$ 、 $k_y$ , and $k_z$ are the thermal conductivities of the shell in $x$ , $y$ , and $z$ directions, $H_r$ is the heat released by the final reaction of the resin, ${\alpha }_{\mathrm{cure}}$ is the curing degree,and $\frac{d{\alpha }_{\mathrm{cure}}}{dt}$ is the curing reaction rate of the resin.

${\rho }_c{c}_{pc}\frac{\partial T}{\partial t}=\frac{\partial }{\partial x}\left(k_x\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k_y\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left(k_z\frac{\partial T}{\partial z}\right)+\dot{q}$

(10)

where $t$ is time and $\dot{q}$ is the internal heat source. Its value is not constant, and it will change with the curing reaction rate of the resin.

$\dot{q}={{\rho }_{r}H}_u\frac{d{\alpha }_{\mathrm{cure}}}{dt}$

(11)

where $H_u$ is the heat released by unit mass of resin matrix when the curing reaction of resin matrix is completed and ${\rho }_r$ is the matrix density.

Curing kinetics model

The curing reaction rate $\frac{d{\alpha }_{\mathrm{cure}}}{dt}$ per unit time can be obtained by solving the curing kinetic model, so that the instantaneous heat release of the corresponding unit volume can be calculated.

$\frac{d{\alpha }_{\mathrm{cure}}}{dt}=\left(k_1+k_2{{\alpha }_{\mathrm{cure}}}^m\right)\left(1-{{\alpha }_{\mathrm{cure}}}^n\right)$

(12)

In the formula, $k_1$ and $k_2$ are the curing reaction rate constants, which can be expressed by Arrhenius equations.

$k_1=A_1\hspace{0.17em}\exp \left(\frac{-∆E_1}{RT}\right)$

(13)

$k_2=A_2\hspace{0.17em}\exp \left(\frac{-∆E_2}{RT}\right)$

(14)

where $T$ represents the temperature of the composite material, $R$ represents the universal gas constant, and $∆E$ and $A$ represent, respectively, the activation energy and frequency factor, which are constants that can be obtained by experiments.

Establishment of tension system

Equivalent method of interference fit

The key to analyze the winding tension in this paper is to equate the winding with applying tension process as a thin-layer interference fit method in order to convert the nonlinear tension solution problem into a linear displacement–stress calculation problem and simplify the calculation process. The interference fit process is shown in Figure 10. In the winding stage, it is easy to change the state of winding fibers with elasticity since the wound layer can be regarded as having elasticity with flexibility, and the adjacent layers will generate a corresponding radial displacement caused by the radial compressive stress and thermal stress, which is considered to be the interference value. The interference values are relaxation for the internal fiber layers that have been wound. With the increase of the number of the external winding layers, the radial relaxation effect of each winding layer is obviously intensified, the sum of radial displacement increases, and the sum of radial residual stress also increases.

Figure 10.

Schematic of interference fit.

The radial stress value at the inner diameter of the last winding layer is only related to the winding of $\left(i\right)\text{th}$ layer which can be regarded as the research focus for convenient analysis. The relationship between the $\left(i-1\right)\text{th}$ layer and $\left(i\right)\text{th}$ layer can be equivalent to the result of interference fit after winding the $\left(i\right)\text{th}$ layer. For the outer diameter of the $\left(i-1\right)\text{th}$ layer, the radial displacement changes $\mathit{\Delta }{R}_{i-1}$ since the inward extrusion of the $\left(i\right)\text{th}$ layer. For the inner diameter of the $\left(i\right)\text{th}$ layer, the radial displacement changes $\mathit{\Delta }{r}_i$ since it is subjected to the reverse force of the $\left(i-1\right)\text{th}$ layer. ²⁵ Therefore, the displacement at the inner diameter of the outer layer can be obtained according to equation (15).

$u\left(r_i\right)=\mathit{\Delta }{r}_i=\frac{r_i}{E_{\mathrm{h}}}\left[\lambda \left(\frac{{r_i}^{2\lambda }+{R_i}^{2\lambda }}{{r_i}^{2\lambda }-{R_i}^{2\lambda }}\right)-{\mu }_{\theta }\right]*P_{r_i}$

(15)

Thus, the radial displacement $u\left(r_i\right)$ distributed after the interference fit of the $\left(i\right)\text{th}$ layer can be obtained.

When the two winding layers have interference fit, the outer diameter of the inner layer displacement formula can be described as follows:

$u\left(R_{\left(i-1\right)}\right)=\mathit{\Delta }{R}_{\left(i-1\right)}=\frac{R_{\left(i-1\right)}}{E_{\mathrm{h}}}\left[\lambda \left(\frac{{r_{\left(i-1\right)}}^{2\lambda }+{R_{\left(i-1\right)}}^{2\lambda }}{{r_{\left(i-1\right)}}^{2\lambda }-{R_{\left(i-1\right)}}^{2\lambda }}\right)-{\mu }_{\theta }\right]*P_{R_{\left(i-1\right)}}$

(16)

The radial displacement $u\left(R_{\left(i-1\right)}\right)$ distributed after the interference fit of the $i-1$ layer can be obtained according to equation (16).

The complete interference value can be synthesized by $u\left(r_i\right)$ and $u\left(R_{\left(i-1\right)}\right)$ :

${\delta }_i=u\left(r_i\right)-u\left(R_{\left(i-1\right)}\right)$

(17)

where $r_{\left(i-1\right)}$ is the inner diameter of inner layer, $R_{\left(i-1\right)}$ is the outer diameter of inner layer, $r_i$ is the inner diameter of outer layer, $R_i$ is the outer diameter of outer layer, $\lambda$ is the flexibility ratio, ${\mu }_{\theta }$ is the Poisson coefficient, $E_h$ is the elastic modulus of resin matrix, and ${\hspace{0.17em}P}_i$ is the surface pressure of winding layer.

$\lambda =\frac{Ec·h_c}{E_M·h_M}$

(18)

where $E_M$ is the elastic modulus of mandrel, $E_c$ is the elastic modulus of composite, $h_c$ is the thickness of composite material, and $h_{M\hspace{0.17em}}$ is the thickness of mandrel.

The radial displacement produced by the interference ${\delta }_i$ will cause the winding layer to deform in the radial direction. Therefore, it is necessary to ensure that each adjacent layer produces the equivalent interference value to keep the overall force of the winding layer in balance. It means that the adjacent layers are in a state of equal residual stress. The relationship between the pressure on the surface of each winding layer can be obtained according to the overall balance condition of the composite winding layer. Obviously, the surface pressure of each winding layer $P_{i\hspace{0.17em}}$ can be obtained in the case of a gradient change in tension by equation (19). The surface pressure of each winding layer ²⁶ can be expressed as follows:

$P_i=\frac{{\sigma }_i\left(r_i-R_{\left(i-1\right)}\right)}{\left(r_i+R_{\left(i-1\right)}\right)/2}$

(19)

The given hoop stress ${\sigma }_{\theta }$ should be transformed into the corresponding radial ones ${\sigma }_i$ through hoop-radial stress balance formulas if the given values are hoop stress.

For the $\left(n\right)th$ layer, ²⁷ the corresponding radial pre-stress ${\sigma }_{in}$ can be solved by equations (20) and (21).

${\sigma }_{in}=\frac{t_c{\sigma }_{\theta n}}{r_n^{\prime }}$

(20)

${\sigma }_{\theta n}=\frac{1}{t_c}·\int {\sigma }_{\theta n}\left(r\right)dr$

(21)

where $r_n^{\prime }$ is the inner diameter value of the $\left(n\right)th$ layer.

Model of tension system

It can be seen from equation (19) that the radial pre-stress ${\sigma }_i$ is proportional to the surface pressure of the winding layer ${\hspace{0.17em}P}_i$ . The radial pre-stress of the outer winding layer will decrease with the increase of winding layers, so the radial stress of the outer layer is smaller than that of the inner layer. It will intensify the radial relaxation effect between adjacent layers if the equal tension winding system is still used, resulting in serious curing delamination in the molded flywheel. Therefore, according to the actual radial stress of each winding layer, it is more reasonable to adopt a variable tension system to apply tension to the winding layer in a gradient decreasing manner. The size of mandrel is converted into fiber equivalent to facilitate analysis and calculation, and the winding tension formula can be expressed as follows:

$F_i=\frac{{\delta }_M}{{\delta }_1}·\frac{Ec·\delta c+E_M·{\delta }_M}{\frac{\left(i-1\right)}{N}{E}_c·{\delta }_c+E_M·{\delta }_M}·{\sigma }_i$

(22)

where $F_i$ is the winding tension, N is the number of winding layers, $i$ is the current winding layer, ${\delta }_M$ is the thickness of the mandrel, and ${\delta }_1$ is the interference value between the first winding layer and the metal mandrel.

In the single winding stage of two-step molding process, the radial pre-stress of each winding layer is mainly the equivalent residual stress of the interference fit between adjacent layers. However, the original stress balance between the winding layers will be destroyed since the coupling effect of temperature-curing degree on various characteristics of the composite materials, so the radial residual total stress ${\sigma }_{\mathrm{total}}$ will also change in the heated-mandrel winding process. It is mainly composed of the thermal stress ${\sigma }_i^{ch}$ caused by temperature gradient during curing, curing shrinkage stress ${\sigma }_i^{th}$ caused by curing reaction, and external compressive stress ${\sigma }_i^T$ caused by winding tension component during winding. According to the different directions of each stress, the stress balance relationship of the winding layers is shown in Figure 11.

${\sigma }_{\mathrm{total}}={\sigma }_i^{ch}+{\sigma }_i^{th}+{\sigma }_i^T$

(23)

Figure 11.

Stress balance relationship of the winding layers.

Finally, equation (23) can be substituted into equation (22) to replace the value of ${\sigma }_i$ with ${\sigma }_{\mathrm{total}}$ . Thus, the tension system considering the influence of coupling field based on heated-mandrel winding process can be obtained. The variables ${\sigma }_i^{ch}$ 、 ${\sigma }_i^{th}$ , and ${\sigma }_i^T$ involved in the calculation process can be solved by the following models.

Model of thermal stress

In the heated-mandrel winding process, the heat is transferred from the inner layers to the outer layers as the curing temperature increases. The temperature difference between the inner and outer layers creates a temperature gradient, which generates thermal stress for outward expansion. $\mathit{\Delta }{T}_i$ represents the temperature change of each winding layer along the radius direction, i = 1, 2....n, and $\mathit{\Delta }{\epsilon }_i^{ch}$ is the thermal strain. Then the thermal stress ${\sigma }_i^{ch}$ of each layer can be expressed as follows: ²⁸

${\sigma }_i^{ch}=\mathit{\Delta }{\epsilon }_i^{ch}*E_c=\left(\gamma ·\mathit{\Delta }{T}_i\right)*E_c$

(24)

$\mathit{\Delta }{\epsilon }_i^{ch}=\gamma ·\mathit{\Delta }{T}_i$

(25)

Model of curing shrinkage stress

In the heating–curing process, the volume shrinkage of composite will generate the inward curing shrinkage stress in the curing reaction. ²⁸ An arbitrary cube micro-unit is selected on a certain winding layer of the composite. The status of this micro-unit after the curing shrinkage reaction can be depicted, and the strains of this micro-unit along three directions are ${\epsilon }_1$ 、 ${\epsilon }_2$ , and ${\epsilon }_3$ , respectively. Thus, the curing shrinkage value $∆v$ can be calculated as follows:

$\mathit{\Delta }v=\frac{dv-d{v}_0}{d{v}_0}=\left(1+{\epsilon }_1\right)\left(1+{\epsilon }_2\right)\left(1+{\epsilon }_3\right)-1$

(26)

Because the resin is a kind of isotropic material, the curing shrinkage stress in each direction and the strains are equivalent, that is, ${\epsilon }_1={\epsilon }_2={\epsilon }_3$ . The curing shrinkage strain $\mathit{\Delta }{\epsilon }_i^{th}$ of each layer of the composite can be expressed as follows: ²⁹

$\mathit{\Delta }{\epsilon }_i^{th}=\sqrt[3]{\left(1+\mathit{\Delta }{\nu }_i\right)}-1$

(27)

where $\mathit{\Delta }{\nu }_i$ is the chemical strain value of the composite, $∆{\alpha }_i$ is the change in the curing degree with the temperature gradient, and the linear change relationship between $\mathit{\Delta }{\nu }_i$ and $∆{\alpha }_i$ is $∆v_i=\beta ∆{\alpha }_i$ , where $\beta$ is the volume shrinkage ratio. Therefore, the curing shrinkage stress ${\sigma }_i^{th}$ caused by the curing reaction can be expressed as follows:

${\sigma }_i^{th}=\mathit{\Delta }{\epsilon }_i^{th}*E_c=\left(\sqrt[3]{\left(1+\beta \mathit{\Delta }{\alpha }_i\right)}-1\right)*E_c$

(28)

Model of radial external pressure stress

The radial component of tension applied to the winding layer is also one of the main components of radial residual stress. In fact, the radial component of winding tension is the radial external pressure stress on each winding layer considering the actual stress relationship of each layer in the winding with applying tension process. ³⁰ The corresponding radial external compressive stress ${\sigma }_i^T$ can be expressed as follows:

${\sigma }_i^T=\frac{F_i}{R_i}=f\left({\sigma }_i\right)$

(29)

Numerical simulation and analysis

Establishment of finite element model

Since the mandrel is in direct contact with the innermost layer of the composite in finite element model, the mandrel is converted into fiber equivalent and simplified as a part of the composite tube to reduce the analysis difficulty and ensure the rationality of the analysis. ³¹ The mandrel is #45 steel which is composed of a hollow cylindrical metal mandrel and two hoop metal disks at both ends. The inner diameter is 20 mm, wall thickness is 10 mm, and length is 200 mm. The inner hoop radius of the hoop-shaped disks is 20 mm and outer hoop radius is 60 mm. The thickness is 6 mm. The composite tube is a hollow cylinder which is composed of carbon fiber and Bismaleimide (BMI) 5428. The inner diameter is 30 mm, wall thickness is 20 mm, and length is 200 mm. The finite element models are shown in Figure 12.

Figure 12.

Finite element model of composite and mandrel.

The curing process will undergo a state transition from flow state, viscous-flow state, viscoelastic state, and glass state for thermosetting resin matrix. The boundary between different states is mainly marked by the change of physical parameters such as curing degree, elastic modulus, and thermal expansion coefficient of the resin matrix. Therefore, this paper only considers the changes of physical parameters caused by the changes of state and ignores the change of physical parameters caused by other factors in the same state to ensure the stability and convergence of numerical simulations.^32–36 The physical parameters are shown in Table 2, and the curing kinetics parameters are shown in Table 3.

Table 2.

Equivalent physical performance parameters of composite.^15,23

Component	Density (Kg/m3)	Specific heat (J/Kg $·℃$ )	Thermal conductivity (W/m $·℃$ )
Composite	1760	2639.5	0.89

Table 3.

Kinetic parameters of composite.^15,23

$m$	$n$	$l$	$A1\hspace{0.17em}\left(s^{-1}\right)$	$A2\left(s^{-1}\right)$	$∆E_1\hspace{0.17em}\left(J/mol\right)$	$∆E_2\left(J/mol\right)$	$Hr\hspace{0.17em}\left(J/Kg\right)$
1.421	1.456	2.76	6.73 × 106	3.82 × 106	8.864 × 104	7.736 × 104	4.43 × 105

Equivalent density:

${\rho }_c=f{\rho }_f+\left(1-f\right){\rho }_r$

(30)

Equivalent specific heat:

${\rho }_{pc}=\frac{f{\rho }_f{c}_f+\left(1-f\right){\rho }_r{c}_r}{p_c}$

(31)

Equivalent conductivity:

$K_c=\frac{K_f{K}_r{\rho }_c}{{fp}_f{K}_f+\left(1-f\right){\rho }_r{K}_r}$

(32)

where $f$ represents fiber and $r$ represents resin.

The curing reaction kinetic equation of BMI 5428 is as follows:

$\frac{d{\alpha }_{\mathrm{cure}}}{dt}=A_1*\hspace{0.17em}{\exp }^{\frac{{-∆E}_1}{RT}}\hspace{0.17em}*{\left(1-{\alpha }_{\mathrm{cure}}\right)}^l+A_2*\hspace{0.17em}{\exp }^{\frac{{-∆E}_2}{RT}}{\left(1-{\alpha }_{\mathrm{cure}}\right)}^n*{{\alpha }_{\mathrm{cure}}}^m$

(33)

Thermal analysis process

The temperature and curing degree obtained by thermal analysis will directly affect the physical parameters of the composite materials and then affect the structural analysis results, as shown in Figure 13. Therefore, the accuracy of the thermal analysis results is the premise to ensure that the composite materials can successfully complete the winding–curing process and become a high-quality flywheel product. The thermal analysis process is described in Figure 14.

Figure 13.

Thermal analysis diagram.

Figure 14.

Flow chart of thermal analysis.

Firstly, the parameters required for calculation are initialized and the boundary conditions such as temperature load are applied. Then the thermal analysis equation based on the initial load step is solved, and the obtained temperature results are compared with the initial curing temperature. If the condition $T>T_{\alpha cure}$ is satisfied, the comparison result will be returned to the thermal analysis equation to calculate the temperature of the next load step. At the same time, the corresponding curing degree value is calculated by combining the kinetic parameters, and the curing degree is judged continuously. The cycle will stop until the curing degree is close to 1, that is, the complete curing state is reached, and the internal cycle is ended. The temperature load is loaded completely or will not be further judged. The program cycle will be ended until all the temperature loads are loaded, and the data results of temperature and curing degree will be saved to call the data for structural analysis.

Structural analysis process

The structural analysis of this paper is based on the calculation results of the thermal parameters of the composite materials obtained by the above thermal analysis process. It includes the equivalent thermal expansion coefficient, elastic modulus, thermal stress, and curing shrinkage stress. Meanwhile, the changes of these thermal parameters will react on the corresponding physical parameters in the thermodynamic model. Therefore, there is a close coupling relationship between the structural analysis and thermal analysis. The structural analysis diagram is shown in Figure 15.

Figure 15.

Structural analysis diagram.

The structural analysis process is described in Figure 16. Firstly, the parameters required for calculation are initialized and the temperature and curing degree obtained from thermal analysis are taken as the boundary conditions for structural analysis. Whether the curing degree of the current load step reaches the gel curing degree will be judged. If not, add a load step, update the process conditions, and recalculate the curing degree of the current step until the curing degree reaches the gel curing degree. If the temperature reaches the gel temperature, it is necessary to further judge whether the glass transition temperature can be satisfied. If it is satisfied, the thermal stress model and the curing shrinkage stress model will be called to solve the thermal stress and curing shrinkage stress. If it is not satisfied, the mechanical parameters will be re-updated until the conditions are satisfied. Finally, the whole heating process is completed, and the structural analysis process is terminated.

Figure 16.

Flow chart of structural analysis.

Numerical simulation analysis

The relevant research in the later part of this paper assumes that the mandrel size is converted into fiber equivalent to simplify the analysis and calculation process. According to the research method of layer-by-layer displacement coupling theory based on temperature parameters in Reference. 37, each winding layer will undergo a curing reaction in turn under the action of the applied temperature parameter variables. As a result, the original stress balance between the adjacent winding layers is broken. In the analysis process, the numerical simulation of winding tension needs to be assumed based on the following descriptions.

(1) Neglect the layer thickness changes caused by resin flow during winding. The resin distribution in each layer is uniform, and the fiber is in elastic range.

(2) The plane stress state is mainly studied. ³⁸ The interference fit occurs in the radial direction and the axial stress component is ignored.

(3) There is a hard contact between the metal mandrel and the first winding layer, and the binding constraint $tie$ is adopted. The radial displacement is continuous, and there is no relative sliding in the hoop direction.

The simulation process of winding tension is shown in Figure 17.

Figure 17.

Analysis process of tension system.

Firstly, input the initial conditions such as the number of winding layers, material thickness, and initial pre-stress for winding. The winding starts from the first layer and applies binding constraint $tie$ to the first layer and the metal mandrel, making them to be a complete structure. Then, calculate the inner and outer radius values of each winding layer based on the initial winding data, and calculate the radial interference of each winding layer ${\delta }_i$ . It is judged whether each ${\delta }_i$ is equal according to the stress balance condition that satisfies the equal radial displacement of adjacent winding layers. If not equal, return to the previous step to recalculate the interference. If equal, the outer surface pressure of each winding layer is calculated according to the functional relationship between $P_i$ and radius values. Then the pre-stress ${\sigma }_i$ of the adjacent layers will be derived from the relationship between the pressure and pre-stress of the fiber layer. It is judged whether the variation of residual stress ${∆\sigma }_{total}$ is small enough. It satisfies the condition of equal residual stress if it is small enough and will be substituted into the tension solution model to develop a reasonable tension system. Otherwise, the residual stress ${∆\sigma }_{total}$ will be directly controlled by adjusting the curing system, or it will be indirectly controlled by adjusting the initial pre-stress by compensating the pre-stress deviation value ${∆\sigma }_i$ until the condition ${∆\sigma }_{total}\approx \epsilon$ is satisfied.

In the heated-mandrel winding process, the ANSYS temperature-curing degree file is imported into the structural analysis to analyze the influence of coupling field changes on thermal parameters and the variation law of radial residual stress between adjacent winding layers affected by the coupling field results based on the stress balance formula of the winding layer and the relationship between winding tension and pre-stress. The relationship between the variation law of radial residual stress and the curing system is summarized, which is beneficial to obtain the optimal winding tension system as shown in Table 4.

Table 4.

Relationship between coupling field and residual stress.

Time/s	Temperature contour	$\mathit{T}$ /°C	Curing degree contour	$\mathit{\alpha }$	Layer	Pre-stress/MPa	Residual Stress/MPa
100					1	322.5	46.7
					2	307	46.2
					3	294.1	45.6
					4	283.3	44.8
					5	273.8	44.1
					6	265.8	43.5
300					1	329.4	47.8
					2	320.7	47.1
					3	315.3	46.9
					4	294.2	46.1
					5	285.1	45.4
					6	273.6	44.6
500					1	339.6	48.7
					2	325.8	48
					3	313.6	47.2
					4	302.4	46.3
					5	292.8	45.5
					6	285	44.7
700					1	346.6	49.5
					2	335.5	48.7
					3	330.8	47.6
					4	323.6	46.8
					5	305.1	45.7
					6	291.3	44.8
900					1	358.3	50.3
					2	343.9	50.1
					3	332.8	50.4
					4	321.4	50.2
					5	310.2	50.3
					6	302.3	50.2
1100					1	368.2	51.4
					2	360.8	50.6
					3	352.6	50
					4	343.7	49.4
					5	328.5	48.7
					6	306.3	48.1
1300					1	378.5	52.3
					2	364.2	51.2
					3	351.5	50.3
					4	340	49.3
					5	329.7	48.5
					6	317.3	47.6
1500					1	380.3	52.7
					2	371.5	51.9
					3	362.3	51.2
					4	351.6	50.5
					5	340.2	49.7
					6	329.5	48.6
1700					1	390.5	51.2
					2	378.2	50.7
					3	369.4	50.1
					4	357.6	49.5
					5	351.8	48.9
					6	340.7	48.2
1900					1	395.8	51.7
					2	385.3	50.9
					3	376.2	50.3
					4	363.7	49.6
					5	355.8	49.1
					6	346.3	48.5

The initial heating temperature of the inner surface of the mandrel is 280°C. Seen from the temperature contour, the heating rate of the hoop-shaped metal disks at both ends is faster than that of the composite materials in the initial heating stage since the thermal conductivity of the metal is higher than that of composite materials. When the heating time reaches about 900 s, the heating rate inside the composite materials is significantly accelerated. When the heating continues to about 1100 s, a large amount of heat is released due to the continuous curing reaction inside the composite. Therefore, although the temperature of hoop-shaped metal disks is still increasing, the heating rate is not as fast as the heat release rate of the curing reaction inside the composite materials, so a large amount of heat will make the internal temperature of the composite begin to be higher than that of the hoop-shaped metal disks at both ends. When the time reaches 1300s, the high temperature area is further expanded inside the composite materials, and it will gradually diffuse from the inside to the outside with time and finally disappear with the completion of curing. Seen from the curing contour, the innermost layer of the composite has the fastest heating rate and the earliest curing time since it is in direct contact with the surface of the heated metal mandrel. According to the thermal conductivity characteristics of the metal mandrel, the direction of curing reaction is from the inside to the outside, and from both ends to the middle. Before 800s, the resin matrix is in a low modulus and uncured viscous flow state. The curing reaction rate is slow, and the curing degree is less than 30%. Between 800s and 1400s, a part of resin begins to change into a viscoelastic state with high modulus, and the curing reaction rate is accelerated. The resin is currently in a mixed viscous flow-viscoelastic state, and the curing degree is close to 60%. After 1500s, the curing reaction will release more heat as an internal heat source with the aggravation of the curing degree of the resin matrix, which will further increase the temperature and curing rate. The curing range will exceed 90%, and the whole curing process is basically completed. The temperature change process and the curing degree change process over time at different positions are shown separately in Figures 18 and 19.

Figure 18.

Temperature change process over time.

Figure 19.

Curing degree change process over time.

The residual stress between the adjacent winding layers will increase with the increase of the curing temperature, and the radial residual stress of each winding layer still follows the law of decreasing layer by layer from the inside to outside. The radial stress distribution in different curing times is shown in Figure 20.

Figure 20.

Radial stress distribution in different curing times.

It is not difficult to see from Table 4 that the regularity of temperature and curing degree obtained by the simulation contours is consistent with the theoretical analysis results in the curing system of this paper, which meets the curing expectation. However, the radial residual stress between adjacent winding layers is quite different, so that the stress state between adjacent layers is difficult to balance. Therefore, it is necessary to readjust the pre-stress values to compensate the deviation of residual stress ${∆\sigma }_i$ to reduce the residual stress ${∆\sigma }_{total}$ and make the residual stress between adjacent winding layers equal under the premise that the curing system remains unchanged. Therefore, different pre-set stress values are brought into the tension system solution model to calculate the pre-stress of each winding layer. Then the pre-stress values are brought into the residual stress formulas to calculate the residual stress. Finally, the overall deviation range of the residual stress of each layer can be obtained. The 6-layer winding is taken as an example to solve eight times and the results are compared. Finally, the range of pre-stress which can make the stress balance between adjacent layers is found, as shown in Table 5.

Table 5.

Relationship between residual stress deviation and pre-stress.

Times	Layers
	Pre-stress
	1	2	3	4	5	6	Residual stress deviation (%)
1st	586.5	567.4	556.2	547.5	536.2	523.7	25
2nd	537.9	523.4	517.6	503.8	485.5	476.4	20
3rd	459.1	437.4	421.6	413.9	403.5	389.1	17
4th	404.3	385.6	376.2	363.7	351.8	347.3	11
5th	368.5	358.2	346.5	337.2	324.2	306.4	3
6th	390.6	372.3	368.5	353.2	337.1	325.6	7
7th	322.5	314.7	308.3	285.9	273.8	262.5	15
8th	291.5	273.4	254.1	239.5	223.7	219.6	19

When the pre-stress is set between 300 MPa and 370 MPa, it is found that the corresponding residual stress deviation range is the smallest after being substituted into the solution model. The residual stress between adjacent layers is close to the same state, and the stress state is basically balanced. The wound fiber layer has higher compactness ³⁹ and better compression resistance after molding, as shown in Figure 21.

Figure 21.

Relationship between residual stress deviation range and pre-stress.

Verification of the tension system model

The molding of composite flywheel based on the heated-mandrel winding process is taken as an example to study the applicability of the tension system model established in this paper. At the same time, it also studies how to achieve the stress balance in the radial direction of each winding layer by adjusting the pre-stress or compensating the radial residual stress to ensure that the radial residual stress of adjacent layers is equal.

In the experiment, a stress detector LM-12 is selected to measure the residual stress. The LM-12 has the characteristics of high sensitivity and accuracy. Most importantly, its stability and anti-interference performance are better while measuring the residual stress. After molding of a thick-walled composite flywheel, we recorded the residual stress data captured with LM-12. The experiment steps designed in this paper are shown as follows.

(1) Firstly, the optimal curing system (Table 4) and the radial pre-stress (Table 5) are determined according to the numerical simulation results of the model, and then the theoretical radial residual stress of each winding layer is calculated according to the solution models of the tension system in this paper.

(2) Then, the actual radial residual stress values collected by the Bragg fiber grating sensor in real time are compared with the values calculated by the formula to verify the reliability of the theoretical model.

(3) Finally, the radial residual stress in the molding flywheel is tested using the split method. An integral portion is cut out as an experimental sample, and it is made to release all the residual stress layer by layer. The released stress of each layer can be measured using LM-12, where the relative difference is used to compare the degree of proximity between the calculated values and the experimental data to verify whether the tension system established this paper can achieve the purpose of equal radial residual stress existing in adjacent layers after winding.

Taking the heated-mandrel winding process as the research object, the radial residual stress is tested. Four groups of pre-stress with relatively small residual stress deviation in Table 5 are taken into the tension solution model, respectively. The transverse comparison between the calculated residual stress results and the measured residual stress results in the same process is carried out. The calculation formula of relative difference $∆{\sigma }_i=\left[{\sigma }_{i\left(cal\right)}-\left({\sigma }_{i\left(ex\right)}\right)/{\sigma }_{i\left(cal\right)}\right]·100\%$ is introduced to describe the agreement between the radial stress measurement results and the calculation results to verify the accuracy of the established tension model in this paper.

It can be seen from Table 6 that when the pre-stress is $\sigma$ 4, the calculated values and the measured values of the residual stress have better tracking characteristics, the values are the closest, and the relative differences of radial residual stress are also the smallest. The results shown in Figure 22 are consistent with the conclusion in Table 6. Therefore, $\sigma$ 4 is the most reasonable winding pre-stress in the heated-mandrel winding process after the curing system is determined. And it is not difficult to find that $F_4$ is the optimal tension system based on the process after $\sigma$ 4 is substituted into the tension solution model.

Table 6.

Comparison of residual stress corresponding to different pre-stress. ²⁶

Winding layers	1	2	3	4	5	6
Pre-stress $\sigma$ 1	322.5	314.7	308.3	285.9	273.8	262.5
Calculated residual stress ${\sigma }_{cal}$ (Mpa)	49.6	50.3	49.4	49.7	49.2	49.1
Experimental residual stress ${\sigma }_{ex}$ (Mpa)	48.2	48.5	48.6	48.3	48.5	48.7
Relative difference $\mathit{\Delta }\sigma \%$	0.3	3.7	1.6	2.9	1.4	0.8
Winding layers	1	2	3	4	5	6
Pre-stress $\sigma$ 2	404.3	385.6	376.2	363.7	351.8	347.3
Calculated residual stress ${\sigma }_{cal}$ (Mpa)	53.6	52.3	52.4	51.6	53.2	54.1
Experimental residual stress ${\sigma }_{ex}$ (Mpa)	51.2	52.4	51.5	52.6	51.4	53.7
Relative difference $\mathit{\Delta }\sigma \%$	4.7	−0.2	1.7	−1.9	3.5	0.7
Winding layers	1	2	3	4	5	6
Pre-stress $\sigma$ 3	390.6	372.3	368.5	353.2	337.1	325.6
Calculated residual stress ${\sigma }_{cal}$ (Mpa)	52.8	52.1	51.5	52.1	51.7	51.4
Experimental residual stress ${\sigma }_{ex}$ (Mpa)	52.5	51.7	51.8	51.3	51.1	51.3
Relative difference $\mathit{\Delta }\sigma \%$	0.6	2.1	−0.6	1.6	1.2	0.2
Winding layers	1	2	3	4	5	6
Pre-stress $\sigma$ 4	368.5	358.2	346.5	337.2	324.2	306.4
Calculated residual stress ${\sigma }_{cal}$ (Mpa)	50.3	50.1	50.7	50.5	50.2	50.6
Experimental residual stress ${\sigma }_{ex}$ (Mpa)	50	49.7	50.4	50.2	49.9	50.2
Relative difference $\mathit{\Delta }\sigma \%$	0.6	0.8	0.6	0.6	0.6	0.8

Figure 22.

Comparison of calculated residual stress and measured ones in different pre-stress.

It can be seen from Figure 23 that four different tension systems can be obtained according to four different radial pre-stress values. Among them, only the tension system $F_4$ corresponding to the pre-stress $\sigma$ 4 presents a uniform and smooth gradient decreasing characteristic of winding tension, which is a typical variable tension winding method and consistent with the theoretical results in Reference 10.

Figure 23.

Comparison of tension system according to different radial pre-stress.

The residual stress between layers in the single winding tension system is much larger than that in the heated-mandrel winding process, so it is difficult to ensure the close bonding between the adjacent layers, which is not suitable for the heated-mandrel winding process under the same initial radial pre-stress condition, as shown in Figure 24.

Figure 24.

Comparison of radial residual stress distribution in different molding processes.

When the winding pre-stress values are the same, compare the calculation results of the single winding process and the heated-mandrel winding process through experiments. The radial residual stress values collected by the sensor for every 2 mm winding thickness are recorded in different molding processes. The actual radial equivalent stress value on the mandrel is collected once after each stage of winding is completed. Then, vertical comparison of radial residual stresses between different processes is conducted from both theoretical calculation and experimental measurement. The formula derivation of the tension model and the actual measurement results are comprehensively compared from different perspectives to verify the applicability of the established tension model, as shown in Table 7.

Table 7.

Comparison of radial residual stress in different molding processes.

Process	Layer
	1	2	3	4	5	6	Calculation results
Single winding	368.5	358.2	346.5	337.2	324.2	306.4	Pre-stress $\sigma$ 4/MPa
	47.7	46.1	44.9	43.8	42.9	42.3	Radial residual stress/MPa
	1.6	2.5	2.1	3.2	2.5	1.8	Relative difference $∆$ %
	15.1	14.7	13.5	12.8	11.6	11.2	Winding tension $F_4$ /N
	322.5	314.7	308.3	285.9	273.8	262.5	Pre-stress ${\sigma }_1$ /MPa
	47.2	45.3	43.5	42.1	40.8	40	Radial residual stress/MPa
	2.4	2.7	3.2	3.5	3.7	2.8	Relative difference $∆$ %
	14.3	13.9	13.1	12.3	11.2	10.8	Winding tension $F_1$ /N
Heated-mandrel winding	368.5	358.2	346.5	337.2	324.2	306.4	Pre-stress ${\sigma }_4$ /MPa
	50.3	50.1	50.7	50.5	50.2	50.6	Radial residual stress/MPa
	0.6	0.8	0.6	0.6	0.6	0.8	Relative difference $∆$ %
	17.2	16.4	15.9	15.3	14.8	14.6	Winding tension $F_4$ /N
	322.5	314.7	308.3	285.9	273.8	262.5	Pre-stress ${\sigma }_1$ /MPa
	49.6	50.3	49.4	49.7	49.2	49.1	Radial residual stress/MPa
	0.3	3.7	1.6	2.9	1.4	0.8	Relative difference $∆$ %
	16.5	15.8	15.8	15.1	14.6	13.9	Winding tension $F_1$ /N

Therefore, the formulation of the tension system needs to consider the influence of the temperature field on the thermal parameters to establish a thermal-dependent tension field solution model considering the effect of temperature-curing degree coupling in the heated-mandrel winding process. The imbalance of residual stress between adjacent winding layers is compensated by adjusting the pre-stress after the reasonable curing system is established. Since the radial residual stress between adjacent winding layers has the characteristic of decreasing layer by layer from inside to outside, the variable tension system adopted in this paper can compensate the imbalance of residual stress between adjacent layers more effectively by gradient decreasing winding with tensioning compared with the equal tension system in Reference. 8 and the equal torque system in Reference. 10, and finally achieve the purpose of radial equal residual stress. The comparison results of winding tension and radial residual stress in different systems are shown in Figures 25 and 26, respectively.

Figure 25.

Comparison of winding tension in different systems.

Figure 26.

Comparison of radial residual stress in different systems.

Conclusions

In this paper, the molding process of the composite flywheel and the establishment of the winding tension system are studied based on the heated-mandrel winding process. Considering the influence of the coupling field on the thermal parameters, the tension calculation formula is re-derived, and the thermal-dependent tension field solution model considering the coupling effect of temperature-curing degree is established.

(1) The optimal curing system is determined by using ANSYS numerical simulation. On this basis, the radial residual stress is compensated by adjusting the winding pre-stress, so that the adjacent winding layers of the molded flywheel are in the state of equal residual stress, which makes the winding layers achieve stress balance in the radial direction. Thus, the optimal winding tension system can be obtained by the numerical calculation method in this paper.

(2) The different pre-stress values are substituted into the same calculation model to test the residual stress in the same molding process. The degree of agreement can be described by the transverse comparison between the calculated residual stress results and the measured results to verify the accuracy of the established tension model.

(3) The comparison test of the radial residual stress between two-step molding process and heated-mandrel winding process is carried out. The vertical comparison of the formula-calculation results and the actual-measurement results of the winding tension are comprehensively compared from different perspectives to verify the applicability of the established tension model.

Both the numerical simulation results and experimental results show that the tension applied to each winding layer according to the tension system established in this paper can ensure that each adjacent winding layer bears equal residual stress in the radial direction in the heated-mandrel winding process. Compared with the equal tension system and equal torque system, the tension system established in this paper is more suitable for the heated-mandrel winding process, and the algorithm is simple and reliable.