**Preparation**: Given the total length *L* of the zero-stress cable, and the cable is divided into n segments. Sufficient accuracy can be achieved if n is not less than 40. When an external force is not on the segment node of the cable, a new node can be added at the point external force applied. The elastic elongation of the cable and the temperature strain are considered in Eqs. (5)–(8).

**Step 1**: Take the tension horizontal component *H*1 = *H*0 and inclination angle θ 1 = θ 0 as the first segment cable, and obtain the angle θi of all cable segments by recursion Eqs. (1) and (3); The calculated coordinates XBC and YBC of point B can be obtained by Eqs. (7) and (8), denoted as XB0 = XBC and YB0 = YBC respectively.

**Step 2**: Calculate the difference (∆X, ∆Y) between the calculated coordinates (XBC,YBC) and the target coordinates (XBT,YBT) of point B by formula (9) and (10). When ∆X and ∆Y meet the accuracy requirements, the form of the cable and the tension of each segment are the target values; otherwise, the following calculations are performed.

**Step 3**: Keeping the inclination angle of the left endpoint A remains the same as in step 1 which is θ 1= θ 0. Giving the tension horizontal component *H*1 = *H*0 a small

increment δ*H* such as δ*H* = 0.0001*H*0, we can get *H*1 = *H*0 + δ*H*. The angles of all cable segments are obtained by recursion Eqs. (1) and (3). The calculated coordinates XBC and YBC of point B can be obtained by formula (7) and (8), denoted by XBH = XBC and YBH = YBC respectively. The influence coefficients of *H* on the coordinates of point B can be obtained:

$$\frac{\partial \mathbf{X}}{\partial H} = \frac{\mathbf{X\_{BH}} - \mathbf{X\_{B0}}}{\delta H} \tag{11}$$

$$\frac{\partial \mathbf{Y}}{\partial H} = \frac{\mathbf{Y\_{BH}} - \mathbf{Y\_{B0}}}{\delta H} \tag{12}$$

**Step 4**: Keeping the tension horizontal component the same as step (1) which is *H*1=*H*0.Giving the inclination angle θ1 = θ0 a small increment δθ such as δθ = 0.0001θ0, we can get θ1 = θ0 + δθ. The angles of all cable segments are obtained by recursion Eqs. (1) and (3). The calculated coordinates XBC and YBC of point B can be obtained by formula (7) and (8), respectively denoted by XBθ = XBC and YBθ = YBC. The influence coefficients of θ on the coordinates of point B can be obtained:

$$\frac{\partial \mathbf{X}}{\partial \theta} = \frac{\mathbf{X}\_{\text{B}\theta} - \mathbf{X}\_{\text{B0}}}{\delta \theta} \tag{13}$$

$$\frac{\partial \mathbf{Y}}{\partial \theta} = \frac{\mathbf{Y}\_{\text{B}\theta} - \mathbf{Y}\_{\text{B0}}}{\delta \theta} \tag{14}$$

**Step 5**: Using the coordinate differences ∆X and ∆Y obtained by Eq. (9)–(10), influence coefficients ∂X ∂ *H*, ∂Y ∂ *H*, ∂X ∂θ and ∂Y ∂θ obtained by Eq. (11)–(14), the corrections of *H*0 and θ0 can be obtained by solving the linear equations:

$$
\Delta H = \left(\Delta \mathbf{X} \frac{\partial \mathbf{Y}}{\partial \theta} - \Delta \mathbf{Y} \frac{\partial \mathbf{X}}{\partial \theta}\right) \Bigg/ \left(\frac{\partial \mathbf{X}}{\partial H} \frac{\partial \mathbf{Y}}{\partial \theta} - \frac{\partial \mathbf{Y}}{\partial H} \frac{\partial \mathbf{X}}{\partial \theta}\right) \tag{15}
$$

$$
\Delta\theta = \left(\Delta\mathbf{X}\frac{\partial\mathbf{Y}}{\partial H} - \Delta\mathbf{Y}\frac{\partial\mathbf{X}}{\partial H}\right) \Bigg/ \left(\frac{\partial\mathbf{X}}{\partial\theta}\frac{\partial\mathbf{Y}}{\partial H} - \frac{\partial\mathbf{Y}}{\partial\theta}\frac{\partial\mathbf{X}}{\partial H}\right) \tag{16}
$$

**Step 6**: Taking *H* = *H*0 + ∆*H* as *H*0 which is the tension horizontal component at the left endpoint of the new iteration, and θ = θ0 + ∆θ as θ0 which is the inclination angle at the left endpoint of the new iteration, the calculation of a new iteration from step 1 is carried out.

When the calculated coordinates of the right endpoint B of the cable in step 2 differ within a certain range from the target coordinates, such as not greater than 0.01 mm, the iteration is completed. The tensions and the form shape of the cable can be obtained using *H*0 and θ 0. The piecewise linear recursion iteration method makes the highly nonlinear problem of the suspension cable structure become a linear problem with only two parameters per step iteration, and the calculation difficulty is significantly reduced. The iterative convergence process can be seen through Eqs. (9) and (10).

### *3.2 Excel Implementation of Iterative Process*

The iterative process can be realized by two Excel tables.

Table 1 the suspension cable is divided into n segments and fill the length *L*i of each segment into column 2. Data in the non-shaded part is filled according to known conditions: vertical force *G*i and horizontal force *F*i of each node are filled into column 3 and column 4. The target coordinate (XBT,YBT) of the right endpoint B, the tensile stiffness EA, and the temperature effect α∆*T* .

The initial tension horizontal component *H*0 and inclination angle θ 0 are given according to "preparation", Data in the shaded part can be calculated automatically by excel using Eqs. (1)–(6). *H*i in column 5 can be calculated by Eq. (1), and θi in column 6 can be obtained by finding the inverse function on both sides of Eq. (3):

$$\theta\_{\rm i+1} = \operatorname{a} \tan \left( \frac{H\_{\rm i}}{H\_{\rm i+1}} \tan \theta\_{\rm i} + \frac{G\_{\rm i}}{H\_{\rm i+1}} \right) \tag{17}$$

∆*x*i in column 7 and ∆*y*i in column 8 can be obtained by Eqs. (5) and (6), the calculated coordinates XBC and YBC of point B can be obtained by Eqs. (7) and (8), and the differences ∆X and ∆Y between the calculated coordinates and the target coordinates of point B can be obtained by Eqs. (9) and (10).

Table 2 the values in the "initial value" row are obtained from Table 1 when the initial values *H*0 and θ 0 were given. Giving *H*0 a small increment δ*H* = 0.0001*H*0, we can get 1.0001*H*0 as the new value of *H*0. When the new value *H*0 is filled in Table 1 in the position *H*0, we can get the row of "influence coefficients of δ*H*" of Table 2 from Table 1. Giving angle θ0 a small increment δθ such as δθ = 0.0001θ0, we can get 1.0001θ0 as the new value of θ0. When the new value θ0 is filled in Table 1

**Table 1** Piecewise linear recursion

**Table 2** Excel iteration calculate

in the position θ 0, we can get the row of "influence coefficients of δθ" of Table 2 from Table 1. According to the data of the shaded part in Table 2, the "iterative correction value" can be obtained by formula (15) and (16) and the "new value" can be obtained for the next iteration calculation

The data in Table 2 can be obtained automatically by calling Table 1. In Table 2, if the initial values *H*0 and θ 0 are given, the new values *H*0 + ∆*H* and θ 0+ ∆θ after iteration can be automatically obtained; Enter *H*0 + ∆*H* and θ 0+ ∆θ as new initial values *H*0 and θ 0 into Table 2 to obtain *H*0 + ∆*H* and θ 0+ ∆θ after a new iteration. Examples show that convergence to the stable value can be achieved after about 3–6 iterations.

### **4 Example**

### *4.1 Example 1*

To verify the feasibility and accuracy of the method, the calculation example is a standard catenary. The cable length is *L* = 120 m and neglecting the elastic extension. The left end point of the cable is A(−50,0), the right end point of the cable is B(50,0), and the cable weight is 11 N/m, as shown in Fig. 2.

The standard catenary equation of is:

$$y = \frac{\cosh(\alpha x)}{\alpha} - \frac{\cosh(50\alpha)}{\alpha} \tag{18}$$

Parameter α is the solution of the length constraint equation:

$$1\sinh(\mathfrak{F}0\alpha) = 60\alpha\tag{19}$$

**Fig. 2** Standard catenary

We can get α = 0.02129737 and the theory value of the sag is *f* t = *y*(0) = 29.23433 m = 29,234.3 mm.

The piecewise linear recursive iterative method is used to solve the problem. The total weight of the cable is G0 = 1320 N, the approximation initial values *H*0 = 0.5G0 = 660 N and θ 0= −30°are selected. Taking n = 40, 60, 80 and 100 for calculation, all the four cases converge to within 0.01 mm after the 5th iteration. The iteration and convergence processes of *H*0 and θ 0 are shown in Table 3, Figs. 3 and 4.

The sag value of the cable calculated by the piecewise linear recursive iterative method is *f* p, and calculate the error of the method is ∆*f* = *f*p − *f*t. we get ∆*f* =

**Table 3** Iterative and convergence process

1.5 mm when n = 100, ∆*f* = 2.4 mm when n = 80, ∆*f* = 4.2 mm when n = 60, and ∆*f* = 9.4 mm when n = 40. This example verifies the convergence speed and accuracy of the method.

### *4.2 Example 2*

The method can be used to calculate the internal force and deformation of suspension bridges under concentrated loads [20]. The suspension cable span is 1000 m, the vector height is 100 m, the section area is A = 0.2 m2, the elastic modulus is E = 2 × 105 N/mm2, and the evenly distributed load along the horizontal direction is 5 KN/m. After the suspension cable reaches self-balance, the horizontal force of the suspension cable and the deformation of the loading point when the concentrated force is applied at the mid-span and 1/4 span positions are calculated respectively.

The total vertical load of the suspension cable before applying concentrated force is G0 = 5 × 1000 = 5000 KN. The cable is divided equally along the horizontal direction and n = 100. After form finding using the method provided in the paper, the tension horizontal component *H*0 = 6249.534 KN, and the inclination angle θ 0 = −21.8014°. The height of the midspan is 100 m, the vertical height of the 1/4 span position is 75 m.

When additional load apply on the mid-span point or 1/4 span, we can carry out the recursion iterative method and calculate the tension horizontal component *H*0 and the inclination angle θ 0 of the left endpoint of the cable. The increment of *H*0 and the vertical displacement of the loading point can be calculated as ∆*H* and ∆*y*.

∆*H*/*H*0 increases with the increase of PK/G0 can be seen in Fig. 5. The incremental ratio of tension horizontal component ∆H/H0 is basically linear with the incremental ratio of load PK/G0, and the slope of secant line to initial tangent line decreases by about 3.3% when PK/G0 = 0.20. Under the same load value, the tension horizontal component increment caused by mid-span load is 32% larger than that caused by 1/ 4 span load.

∆*y*/*f* 0 increases with the increase of PK/G0 can be seen in Fig. 6. There are obvious nonlinearities between the incremental ratio ∆*y*/*f* 0 and the load ratio PK/ G0. The slope ratio of secant line to initial tangent line increases by about 33% when

PK/G0 = 0.20. Under the same load value, the vertical displacement of the loading point in 1/4 span is 38% higher than that in mid-span.

This example proposed to compare to the equations of different scholars to calculate the internal force and vertical displacement of suspension bridge [20]. The formulas are complicated, and the difference between different formulas is large, some results difference between them can reach 15–20% [20]. The method in this paper fully satisfies the balance equation of each node of the cable after divided, and the error is only due to the influence of the curve in a single segment. When n = 100, the calculation error of midspan sag is only 0.005% relative to the theoretical mid-span sag, which fully meets the engineering design requirements.

# **5 Conclusion**

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

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

**Zilong Yang and Fujiang Yang**

**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-bylayer 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.

**Keywords** Thick-walled cylinder · Fiber winding · Residual tension · Prestress

# **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.

Z. Yang (B) · F. Yang

National Key Labortory of Particle Transport and Separation Technology, 300180 Tianjin, China e-mail<EMAIL_ADDRESS>
Research Institute of Physical and Chemical Engineering of Nuclear Industry, 300180 Tianjin, China

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_28

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 multilayer 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.

Elastic modulus *Em*, Poisson's ratio ν*m*, inner radius *rmi*, and outer radius *rmo* are the parameters of a metal liner. Inner radius *rfi*, outer radius *rfo*, radial and circumferential modulus *E*θ and *Er*, Poisson's ratio μθ*r* and μ*r*θ are the parameters of the winding layer.

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

$$\frac{d\Delta\sigma\_r}{dr} + \frac{\Delta\sigma\_r - \Delta\sigma\_\theta}{r} = 0 \tag{1}$$

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

$$\begin{aligned} \Delta \varepsilon\_r(r) &= d \Delta u / dr \\ \Delta \varepsilon\_\theta(r) &= \Delta u / r \\ \Delta \varepsilon\_r &= \Delta \sigma\_r / E\_r - \mu\_{\theta r} \Delta \sigma\_\theta / E\_\theta \\ \Delta \varepsilon\_\theta &= \Delta \sigma\_\theta / E\_\theta - \mu\_{r\theta} \Delta \sigma\_r / E\_r \end{aligned} \tag{2}$$

The border conditions are,

$$\begin{aligned} \Delta \sigma\_r(h) &= -T(h)\Delta h/h\\ \Delta \sigma\_r(r\_i) &= \Delta \sigma\_{rm}(r\_{mo}) \end{aligned} \tag{3}$$

**Fig. 1** Fiber winding geometric model

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).

$$P(h) = 2\lambda r\_{mo}^{\lambda - 1} f(r\_{mo})$$

$$u(r) = \left[ (\lambda - \mu\_{\theta r} - K)(\lambda + \mu\_{\theta r}) \left( \frac{r\_{mo}}{r} \right)^{\lambda} - (\lambda + \mu\_{\theta r} + K)(\lambda - \mu\_{\theta r}) \left( \frac{r}{r\_{mo}} \right)^{\lambda} \right] \frac{r\_{mo}^{\lambda}}{E\_{\theta}} f(r)$$

$$\sigma\_{r}(r) = - \left[ (\lambda - \mu\_{\theta r} - K) \left( \frac{r\_{mo}}{r} \right)^{\lambda + 1} + (\lambda + \mu\_{\theta r} + K) \left( \frac{r}{r\_{mo}} \right)^{\lambda - 1} \right] r\_{mo}^{\lambda - 1} f(r) \tag{4}$$

$$\sigma\_{\theta}(r) = \left[ (\lambda - \mu\_{\theta r} - K) \left( \frac{r\_{mo}}{r} \right)^{\lambda + 1} - (\lambda + \mu\_{\theta r} + K) \left( \frac{r}{r\_{mo}} \right)^{\lambda - 1} \right] \lambda r\_{mo}^{\lambda - 1} f(r) + T(r)$$

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

$$f(r) = \int h^{\lambda} T(h) / [(\lambda + \mu\_{\vartheta r} + K)h^{2\lambda} + (\lambda - \mu\_{\vartheta r} - K)r\_{mo}^{2\lambda}] dh \tag{5}$$

Each coefficient's expression is,

$$K = E\_{\theta} \left[ (1 - \upsilon\_m) + \xi^2 (1 + \upsilon\_m) \right] / E\_m (1 - \xi^2)$$

$$\begin{array}{l} \xi = r\_{mi} / r\_{mo} \\ \lambda = \sqrt{E\_{\theta} / E\_r} \end{array} \tag{6}$$

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.

$$E\_{\theta} = \pi E\_f V\_f / 4V\_a$$

$$E\_r \frac{A\_s \sqrt{V\_a / V\_0} \left[ 5 - \sqrt{V\_f / V\_a} - 4\sqrt{V\_0 / V\_f} \right]}{2\left(\sqrt{V\_a / V\_f} - 1\right)^5} \tag{7}$$

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,

$$
\sigma(r\_i) = \sigma(r\_{i-1}) \quad (i = 2, 3, \dots, N) \tag{8}
$$

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

$$MT = H\tag{9}$$

*T* represents each layer's winding tension.

$$T = \begin{bmatrix} T\_1 \ T\_2 \ \dots \ T\_i \ \dots \ T\_N \end{bmatrix}^T \tag{10}$$

*M* is the coefficient matrix.

⎡⎣*B*1 *B*2 *B*3 ... *Bi* ... *BN*  − *A*1 *B*1 + 1 (*A*2 − *A*1)*B*2*r*λ2 − 1 (*A*2 − *A*1)*B*3 ... (*A*2 − *A*1)*Bi* ... (*A*2 − *A*1)*BN*  ... ... ... ... ... ... −*A j B j* + 1 (*A j* − *A j*−1 )*B j r*λ*j* − 1 ... (*A j* − *A j*−1 ) *Bi* ... (*A j* − *A j*−1 ) *BN*  ... ... ... ... ... − *A N*−2 *B N*−2 + 1 ( *A N*−1 − *AN* −2 ) *B N*−1*r*λ *N*−1 − 1 ( *A N*−1 − *A N*−2 ) *BN*  0 − *A N*−1 *B N*−1 + 1 ( *AN*  − *A N*−1 ) *BN r*λ*N*  − 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (11)

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

The expression of matrix *H* is,

$$H = \begin{bmatrix} \frac{1 - r\_{ml}^2 / r\_{mo}^2}{1 + r\_{ml}^2 / r\_{mo}^2} \frac{\sigma\_0}{2\lambda r\_{mo}^{\lambda - 1}}\\0\\\cdots\\0\end{bmatrix} \tag{12}$$

The definition of Coefficient *A* and *B* is,

$$A\_i = \left[ (\lambda - \mu\_{\theta r} - K) \left( \frac{r\_{mo}}{r\_l} \right)^{\lambda + 1} - (\lambda + \mu\_{\theta r} + K) \left( \frac{r\_l}{r\_{mo}} \right)^{\lambda - 1} \right] \lambda r\_{mo}^{\lambda - 1} \tag{13}$$

$$B\_i = \frac{r\_l - r\_{i-1}}{(\lambda + \mu\_{\theta r} + K) r\_l^{\lambda} + (\lambda - \mu\_{\theta r} - K) r\_{mo}^{\lambda}} r\_i^{\lambda}$$

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,

$$\delta\_i = \frac{T\_{acll}}{T\_{tech}} = \sqrt{\frac{1 + (i - 1)\chi}{1 + n\chi}}\tag{14}$$

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.

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:

(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.

**Fig. 2** Stress distribution of liner and wound layers

**Fig. 3** Relationship between prestress and winding parameters

(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.

We may infer the following from Fig. 4:

**Fig. 4** Influence of fiber volume content on stress

**Table 2** Comparison of winding tension results

### *3.3 Uniform Residual Tension*

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

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.

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.

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.

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

**Table 3** Comparison of experiment parameters

**Fig. 5** Dynamic strain monitoring device

**Fig. 6** Test results of strain

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:

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on the Design and Verification Process of Mechanical Penetrations in Reactor Compartment**

**Qian Zhang, Zuoqin Qian, Qiang Wang, and Xinyu Wang**

**Abstract** Mechanical penetrations, as important pressure pipelines penetrating the reactor compartment, withstand high temperatures and pressures. The current complete design and verification process for mechanical penetrations. This article focuses on the problem of stress concentration and easy damage of the penetration components in the reactor compartment under high temperature and high pressure environment. Combining with the existing regulations of nuclear power plants and ships, finite element analysis method is used to analyze the stress of the penetration components under specific high temperature and high pressure and ship ultimate load coupling. At the same time, based on the simulation analysis results, the structural dimensions of the penetration components are optimized, and a mechanical penetration verification process is designed. The coupled thermal stress results of the penetration indicate that the stress of the penetration is too large at the tail of the sleeve, with the values of primary film stress Pm and primary bending stress Pb being 228.2 and 275.91 MPa, respectively. From this, it can be seemed that there is obvious stress concentration at the junction of the support ring and sleeve, as well as at the transition point of the insulation layer, which is the weakest area of the penetration.

**Keywords** Mechanical penetration parts · Nozzle load · Finite element method · Thermal mechanical coupling analysis

# **1 Introduction**

Floating nuclear power plant (FNPP), also known as floating nuclear power plant, is a mobile small nuclear power plant on the ocean. It combines small nuclear reactors with maritime transportation engineering, which can meet the needs of cooling and heating in urban areas, reducing haze weather, and energy supply in remote areas [1].

Q. Zhang · Z. Qian · Q. Wang · X. Wang (B)

Wuhan University of Technology, Wuhan 430063, China e-mail<EMAIL_ADDRESS>
© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_29

Mechanical penetrations are part of the reactor containment vessel and are components that connect the internal and external pipelines of the containment vessel [2]. According to the relevant requirements of nuclear safety, mechanical penetrations still need to maintain the ability to perform their functions under severe accident conditions [3].

Mechanical penetrations are installed on the cabin walls to transport fluid inside and outside the cabin. As an important component of the cabin structure, they, together with electrical penetrations, shielding doors, and the main structure of the cabin, form the third barrier for radioactive shielding in the cabin [4]. The penetration is connected to the cabin bulkhead and is located in a structurally discontinuous position, where stress is easily concentrated. Therefore, the structural strength design of the penetration should not only ensure the airtightness and integrity of the safety shell, but also meet the mechanical performance requirements under various working conditions. At present, many experts and scholars at home and abroad have conducted research on it. Li [5] used finite element software ABAQUS to simulate the steadystate and transient thermal coupling of pressure vessels, analyzed the law of coupled thermal stress during radial and circumferential changes in the vessel structure, and explored the influence of temperature on heat transfer; Hazizi [6] combined with ASME specifications and designed a cylindrical pressure vessel that will not fail by analyzing a pressurized liquefied petroleum gas vertical pressure vessel with a capacity of 10 cubic meters. Jasion [7] studied a pressure vessel with a special cylindrical shape, established an analytical model of this container composed of a rotating shell, and based on membrane theory, analyzed, and determined the stress state in the container. Chen [8] designed a set of loading test equipment to meet the testing requirements of penetrations under extreme load. It was found that when the ultimate load was applied, the stress was mainly concentrated at the root of the penetration; Tong [9] analyzed the stress and deformation of the containment vessel, as well as its ultimate bearing capacity, using a prestressed finite element model. Song [10] conducted vulnerability analysis and probabilistic safety performance evaluation of the nuclear containment structure under the condition of prestressed tendon fracture based on the finite element model of the nuclear containment. They discussed the changes in concrete compressive stress state and the ultimate bearing capacity of the nuclear containment structure, and analyzed the vulnerability curve and probabilistic safety margin of the nuclear containment structure under the condition of prestressed tendon fracture.

Currently, there are complete regulations in domestic and foreign regulations regarding the evaluation standards for nuclear grade pressure vessels. However, there is no clear regulation on the design verification process and evaluation standards for nuclear grade pressure vessels on ships, such as mechanical penetrations in storage tanks. Therefore, this study combines ships with nuclear power plants. Based on the relevant specifications of nuclear grade pressure vessels and ship structures, combined with theoretical research and specific engineering examples, finite element software is used to conduct thermal coupling simulation analysis on the penetration components of the reactor compartment. Based on the results, the structural dimensions of the penetration components are optimized, and a set of process verification methods is designed. In this paper, a reactor tank penetration with a size of ϕ 273.0 × 25.4 [11] was designed and checked through thermal simulation and stress deformation simulation analysis and calculation.

### **2 Thermodynamic Coupling Theory**

The coupling field can be divided into different physical fields: magnetic field and electromagnetic field, magnetic field and stress field, temperature field and stress field, temperature field and flow field coupling, etc. The coupling methods are divided into direct coupling and sequential coupling. Direct coupling involves introducing coupling effects into the control equation through element matrices or load vectors, and then solving the control equation. The calculation process is convenient but the calculation accuracy is not high, and it is mostly suitable for theoretical analysis of coupled fields; Sequential coupling is the process of solving a single physical field within each incremental step, and then applying the solution of the previous physical field as an external load to the latter physical field to complete the coupling of the two physical fields. Although the calculation process is relatively complex, this method has high computational accuracy and is suitable for engineering calculations of coupled fields [12].

This article adopts the sequential coupling method, and the finite element equation of sequential coupling is:

$$
\begin{bmatrix} K\_{11}X\_1 + K\_{11}X\_2 \ 0 \\ 0 \end{bmatrix} \begin{bmatrix} X\_1 \\ X\_2 \end{bmatrix} = \begin{Bmatrix} F\_1X\_1 + F\_1X\_2 \\ F\_2X\_1 + F\_2X\_2 \end{Bmatrix} \tag{1}
$$

In the formula: *X*1 and *X*2 represent the displacement matrices of two physical fields; *F*1 and *F*2 represent the equivalent load matrices of two physical fields; *K*11 and *K*22 represent the equivalent stiffness matrices when two physical fields exist separately.

Bring the single stress field control equation and the single temperature field control equation into the sequential coupling control equation to obtain the thermal coupling control equation:

$$
\begin{bmatrix} M \ 0 \\ 0 \ 0 \end{bmatrix} \begin{Bmatrix} \ddot{u} \\ \ddot{T} \end{Bmatrix} + \begin{bmatrix} C \ 0 \\ 0 \ C' \end{bmatrix} \begin{Bmatrix} \dot{u} \\ \dot{T} \end{Bmatrix} + \begin{bmatrix} K & 0 \\ 0 \ K' \end{bmatrix} \begin{Bmatrix} u \\ T \end{Bmatrix} = \begin{Bmatrix} F \\ Q + Q^p \end{Bmatrix} \tag{2}
$$

In the formula: *u* and *T* are the element displacement matrix and element temperature matrix, respectively; *C*,*Ct*is the unit damping matrix and the unit specific heat matrix, respectively; *K* , *K t*is the element stiffness matrix and the element heat conduction matrix, respectively; *M* is the unit quality matrix; *F* is the unit load matrix; *Q* is the unit heat generation and unit heat flow rate; *Q p*is the unit plastic heat generation rate.

**Fig. 1** Thermal analysis method process

When the temperature of the penetration changes, deformation will occur, resulting in thermal stress in the structure. The thermal stress of high-energy penetration belongs to secondary stress, which needs to be considered under various working conditions. Therefore, the strength of the penetration must meet the requirements under ultimate load. At the same time, in order to ensure that the temperature of the shielding material is within its working temperature range, thermal coupling analysis of the penetration is necessary. This article uses Ansys software and sequential coupled thermal analysis method to calculate the stress components of each component of the penetration, and verifies the strength of each component of the penetration according to the stress assessment criteria of ASME code [13]. The flow chart of the sequential coupled thermal analysis method in this article is shown in Fig. 1.

### **3 Model and Mesh**

A certain reactor compartment is 1 m × 1m × 1 m enclosed space, pipeline penetrations are key components for fluid transportation inside and outside the reactor compartment, mainly including structural parts and other ancillary structures. The geometric model of the penetration is 1 m × 1m × 1 m 3D model including the penetration. The structure of the penetration includes the middle pressure pipe, the head between the pipe and the sleeve, the sleeve and the support ring; The remaining components are auxiliary structures, including the insulation layer between the sleeve and the penetrating pipe fittings, fixed steel bars, angle steels, and resin plates, radiation resistant materials such as lead plates and lead boron polyethylene, and fire-resistant materials such as polycrystalline wires; The rest is filled with air. The three-dimensional model of the penetration is shown in Fig. 2.

The geometric grid size of the bulkhead and T-shaped support plate is set to 15 mm, the web plate is set to 10 mm, the bar steel, angle steel, and resin plate are set to 5 mm, and the polycrystalline wire is set to 7.5 mm. For penetrations, sleeves, and support rings, the grid size is set to verify the thickness direction of three units. For welding points, the grid size is set to 1 mm, and for other areas, the grid size is

**Fig. 2.** 3D geometric modeling of penetration

set to default. At the same time, select slow transition areas for the grid transition, set the resolution for small areas to 3, and the number of grid cells is 6,865,833. The 3D mesh model is shown in Fig. 3.

**Fig. 3** Mesh division of penetrations

**Fig. 4** Setting of temperature boundary conditions for penetrations

### **4 Boundary Condition**

### *4.1 Temperature Field Boundary Conditions*

The interior of the penetration pipe is a high-temperature and high-pressure fluid. During the steady-state simulation of the temperature field of the penetration and its affiliated structures, the inner wall of the penetration pipe is assumed to be convective heat transfer. When the internal fluid is liquid, the convective heat transfer coefficient is taken as 2000 w/m2 K, and when the internal fluid is gas, the convective heat transfer coefficient is taken as 300 w/m2 K, with an ambient temperature of 25 °C and a convective heat transfer coefficient of 15 w/m2 K; The bulkhead and T-shaped support platform are connected to the storage tank as an insulated wall surface; The contact between other external surfaces and the environment is assumed to be natural convection, and the internal fluid is set as gas. The specific boundary conditions are set as shown in Fig. 4.

### *4.2 Stress Field Boundary Conditions*

The loads borne by the penetration are mainly the design internal pressure, temperature load, and nozzle load. The design internal pressure is set as surface pressure, which acts on the inner wall of the connecting piece along the internal normal direction, at a pressure of 7.0 MPa; There are two commonly used methods for determining the nozzle load of nuclear vessels: ➀ conducting mechanical analysis of the pipeline system to obtain the nozzle load of the vessel; ➁ The allowable load of the nozzle (which is the upper limit value of the nozzle load obtained from the mechanical analysis of the pipeline system) is determined based on the geometric parameters, material characteristics, force torque relationship, and stress assessment criteria under various load conditions of the container nozzle. Method ➁ is not affected by the progress of piping analysis. In the preliminary stage of nuclear vessel design, in order to accelerate the engineering progress, a container-based nozzle load analysis method can be used.

According to the provisions of Section III of the ASME Boiler and Pressure Vessel Code and the Construction Rules for Nuclear Facility Components, the allowable nozzle load of the vessel is calculated to ensure that the maximum stress caused by the design pressure and external load does not exceed the limit of primary stress under design, critical, and accident conditions. Therefore, based on the geometric parameters and material characteristics of the nozzle structure, the stress intensity values at each section of the nozzle can be obtained. Then, according to the overall primary membrane stress intensity Pm, local primary membrane stress intensity PL, Pm (or PL) + primary bending stress intensity Pb stress limit value specified in the ASME code under various working conditions, the allowable nozzle load can be determined by calculating the assumed nozzle load of each type [14]. According to elasticity, the stress calculation of the evaluated section is as follows:

Axial stress:

$$
\sigma\_z = \frac{P}{A} + \frac{p\_i r\_i}{2t} + \frac{M\_{\rm Br} r\_o}{I} \tag{3}
$$

In the formula: *p*i is the internal pressure, Pa; *A* is the cross-sectional area, m2; *r*i is the inner radius, m; *r*o is the outer radius, m; *t* is the wall thickness, m; *I* is the moment of inertia, m4; *M*B is the total bending moment, N m.

Circumferential stress:

$$
\sigma\_0 = \frac{p\_i r\_i}{t} \tag{4}
$$

Radial stress:

$$
\sigma\_r = -\frac{p\_i}{2} \tag{5}
$$

Shear stress:

$$\begin{array}{c} \mathsf{r} = \mathsf{r}\_1 + \mathsf{r}\_2\\ \mathsf{r}\_1 = \frac{\mathsf{V}\mathcal{Q}}{2It}; \mathsf{r}\_2 = \frac{M\_{\mathsf{I}\mathcal{I}^0}}{2I} \end{array} \tag{6}$$

In the formula: Q is the shape coefficient, m3. Section 3 principal stresses σ1, σ2, σ3 is:

$$\begin{aligned} \sigma\_1 &= \frac{\sigma\_\circ + \sigma\_\theta}{2} + \left[ \left( \frac{\sigma\_\circ - \sigma\_\theta}{2} \right)^2 + \pi^2 \right]^{0.5} \\ \sigma\_2 &= \frac{\sigma\_\circ + \sigma\_\theta}{2} - \left[ \left( \frac{\sigma\_\circ - \sigma\_\theta}{2} \right)^2 + \pi^2 \right]^{0.5} \\ \sigma\_3 &= \sigma\_r \end{aligned} \tag{7}$$

The stress difference on the section is:

$$\begin{aligned} S\_{12} &= |\sigma\_1 - \sigma\_2| \\ S\_{23} &= |\sigma\_2 - \sigma\_3| \# \\ S\_{13} &= |\sigma\_3 - \sigma\_1| \end{aligned} \tag{8}$$

The total stress intensity is:

$$S\_{\rm I} = \max(S\_{\rm I2}, S\_{\rm 23}, S\_{\rm 13}) \tag{9}$$

In solid mechanics simulation, the outer wall and connecting plate are set as fixed constraint points, and the nozzle load is calculated according to the formula, which is divided into four types: Tensile force FT = 253.652 kN; Shear force FV = 253.652 kN; torque MV = 126.826 kN m, and Bending moment MB = 126.826 kN m. The torque is clockwise. Due to the connection between both ends of the penetration and the connecting pipeline, four types of nozzles load under normal operating conditions are applied at both ends of the penetration. The loading method is shown in Fig. 5.

### **5 Calculation and Result Analysis**

After setting the boundary conditions, conduct a steady-state analysis and solution of the temperature field of the penetration; Import the temperature field into the mechanical analysis interface for thermal stress analysis, and then load the mechanical boundary conditions again to enter the stress evaluation analysis.

There are two main verification standards for temperature field simulation: 1. Whether the maximum temperature of the lead plate shielding layer exceeds 300 °C; 2. Does the maximum temperature of the lead boron polyethylene shielding layer exceed 120 °C.

For the stress field, set the stress path and extract the membrane stress and bending stress on the stress path. The stress path is shown in Fig. 6.

Calculate the allowable stress of different components according to the specifications, verify each stress component according to the allowable stress, and complete stress assessment. The stress assessment criteria for penetrations are shown in Table 1.

**Fig. 5** Setting of mechanical boundary conditions for penetrations

**Fig. 6** Penetration stress path

**Table 1** ASME evaluation standards

### *5.1 Temperature Field Evaluation*

The temperature field distribution of the penetration after steady-state temperature simulation is shown in Figs. 7, 8 and 9. The overall maximum temperature of the

**Fig. 7** Overall temperature distribution of penetration

penetration is 299.4 °C, and the minimum temperature is 25.151 °C; The maximum temperature of the lead plate shielding layer is 29.103 °C, and the minimum temperature is 26.044 °C; The maximum temperature of the lead boron polyethylene shielding layer is 58.226 °C, and the minimum temperature is 26.157 °C. The calculation results indicate that the temperature evaluation of the penetration meets the design requirements.

### *5.2 Stress Field Evaluation*

The stress field distribution of the penetration after steady-state stress simulation is shown in Fig. 10. The maximum overall stress of the penetration is 651.44 MPa, and the stress is mainly concentrated at the root.

According to ASME Section II specifications, compare the calculated stress components with the corresponding allowable stresses. Under normal working conditions, take a 90° profile stress, and the evaluation results are shown in Table 2. The calculation results show that the stress of the penetration is at paths 11 and 12, with

**Fig. 8** Temperature distribution of lead plate shielding layer

**Fig. 9** Temperature distribution of lead boron polyethylene shielding layer

**Fig. 10** Overall stress distribution of penetrations

the values of primary membrane stress Pm and primary bending stress Pb being 228.2 and 275.91 MPa respectively, which do not meet the evaluation requirement of 207 MPa. The local stress component of the penetration is relatively large, mainly concentrated at the transition between the insulation layer and the support ring of the penetration. The thickness of the sleeve can be increased to strengthen it.

### *5.3 Verification Process for Overall Design of Penetrations*

For the overall design verification of mechanical penetrations, based on the theory of thermal coupling, the ultimate load calculation is combined with the ASME "Boiler and Pressure Vessel Code" Section III and "Nuclear Facility Component Construction Rules". The evaluation criteria are in accordance with ASME Section II specifications, and a complete set of processes is designed. The specific design verification process is shown in Fig. 11.

**Table 2** Profile stress assessment results

**Fig. 11** Verification process for overall design of penetrations

# **6 Conclusion**

This article focuses on the problem of stress concentration and easy damage of the penetration components in the reactor compartment under high temperature and high pressure environment. Combining with the existing regulations of nuclear power plants and ships, finite element analysis method is used to analyze the stress of the penetration components under specific high temperature and high pressure and ship ultimate load coupling. At the same time, based on the simulation analysis results, the structural dimensions of the penetration components are optimized, and a mechanical penetration verification process is designed. Based on the above results, the following conclusions can be drawn:

### **References**

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# **Mechanical Control and Fault Monitoring Analysis**

# **A Machine-Vision-Based Hub Location Detection Technique for Installing Wind Turbines**

**Xuyang Cao, Yongchang Hu, Guoyang Xu, Shuai Song, and Xiaochun Tie**

**Abstract** A position detection approach based on machine vision is developed to address the issue that the traditional wind turbine hub lifting procedure relies on the location of engineers and cannot provide quick feedback on the position information. A camera and a range sensor are mounted on the suspended object to enable real-time detection of the fan hub's posture state in relation to the engine room. A monocular camera positioning technique based on circular features is created for this detection method. The relative pose coordinates of the hub are computed using the position data given by the range sensor. A filter for the range sensor is created based on the hoisting features of the actual working circumstances. The filter applies a filter to the data from the ranging sensor, considerably reducing measurement error and enhancing detection technique stability.

**Keywords** Wind turbine hoisting · Machine vision · Pose detection · Data processing

# **1 Introduction**

The advancement of wind turbine-related technology has raised the bar for crane efficiency, lifting precision, real-time detection, and precise positioning control in wind power construction projects as well as the installation and maintenance of wind turbines. At present, the commonly used domestic wind turbine installation method is to install the nacelle on the tower first, and then the blade and hub are combined on the ground, and the blade and hub assemblies are lifted by large cranes in the high altitude for the installation method. Hub blade assembly in the lifting process, the

X. Cao · Y. Hu (B) · G. Xu

e-mail<EMAIL_ADDRESS>
© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_30

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

S. Song · X. Tie Taiyuan Heavy Industry Co., Ltd., Taiyuan 030024, Shanxi, China

need for multiple cranes to work together, through the rope pulling and other ways to adjust the position of the hub blade assembly.

With the development of wind turbine related technologies, the capacity of wind turbines has been continuously improved, and the installation height of wind turbines has also reached more than 100 m, as shown in Fig. 1, for the installation of wind turbine hubs. Under this working condition, it is difficult to work at high altitude, there are many people involved in installation, and the cost is high. When the hub is installed and aligned, due to factors such as wind power or technical personnel cooperation, the installation period is long and the installation time is greater than 70 h. With the rapid development of China's wind power industry, it is necessary to develop a pose detection method with less direct participation, high safety, high detection efficiency and high detection accuracy.

Compared with other pose detection methods, visual positioning can detect the position of the measured object under non-contact conditions, and can adapt to various complex environments under sufficient illumination. Scholars at home and abroad have accumulated relevant research foundations in machine vision. Liu et al. [1] designed a robot intelligent unstacking system based on visual positioning. The corresponding world coordinates are obtained by using the coordinate transformation of the target pixel center. The position error is 1.1 mm and the angle error is 1.2°. Chen et al. [2] proposed a screw automatic fastening assembly system based on visual positioning. The visual positioning technology based on sub-pixel edge is introduced to accurately compensate the positioning deviation of screw assembly, and the visual positioning accuracy is better than 0.02 mm.

Currently, the detection of camera fixation and the movement of the measured item are the basic foundations of visual positioning research. This study suggests a pose

**Fig. 1** Wind turbine hub lifting

detection technique in which the measured object and the camera are both in a state of erratic motion. This study offers a fan hub based on a circle fitting algorithm that integrates a laser range sensor with machine vision to address the complex working environment.

# **2 The Wheel Hub Pose Detection Scheme Under High Altitude Condition**

Since the characteristics of the cabin itself are difficult to be directly used as positioning features, it is necessary to add feature color blocks as positioning markers, as shown in Fig. 3. The selection of positioning features is the key to the realization of visual detection. For the needs of practical engineering, two square color blocks of a certain size are installed in different directions of the transmission shaft as the positioning feature of the secondary treatment, and the bolt hole of the engine room shaft is used as the feature of the primary treatment.

The camera is used to collect and process the image to obtain the position information of the target point. Combined with the distance information of the ranging sensor, the pixel information on the image is converted into the actual distance information, and the pose state of the fan hub is obtained.

The position of the camera and the ranging sensor is shown in Fig. 2, and the installation position of the ranging sensor and the industrial camera is that the hub is facing the cabin surface. After the camera is installed, the relative position matrix of the camera and the hub center should be obtained to initialize the pose matrix.

a) Camera mounting position b) Industrial camera

**Fig. 2** Sensor mounting

**Fig. 3** Characteristic color block

### **3 Visual Orientation Program**

### *3.1 Image Segmentation Scheme of Secondary Processing*

Since the cabin itself has no obvious features that can be used as features for visual recognition and positioning, it is necessary to add feature color blocks for positioning [3]. The square color block features are prominent, and when performing contour fitting, it can effectively eliminate the influence of the external environment on visual recognition. Therefore, two square color blocks of a certain size are installed in different directions of the drive shaft as the positioning feature of the secondary processing, as shown in Fig. 3. The bolt hole of the nacelle shaft is used as a feature of one-time processing.

The quality of the image the camera captures cannot be guaranteed due to the intricacy of the real operating conditions. As a result, during image processing, secondary processing is used to segment the images [4]. The captured image is first simply processed to separate the main recognition portion from the remainder portion, which is then binarized. The picture is then re-segmented following one procedure: binarization processing is used to produce the desired contour after screening by area, contour, and other parameters [5]. It is a flow chart for secondary processing, as seen in Fig. 4.

This image processing method increases the computational pressure of the host computer to a certain extent and increases the time of image processing, but it can effectively overcome the influence of the environment on image processing and improve the accuracy of visual positioning [6]. In addition, during the installation of

**Fig. 4** Secondary image processing flow

the fan hub, the movement speed is slow, and the influence of the algorithm delay on the installation process can be ignored.

### *3.2 Circle Fitting Algorithm Based on Least Square Method*

When the wheel hub and the engine room are far apart, the area of the positioning color block in the image collected by the camera is small, and it is difficult to extract the features. Aiming at the situation that the color block cannot be identified and located at a long distance, a circle fitting algorithm based on the least square method is introduced [7].

After the image is binarized, the edge of the image will be highlighted. The highlighted pixels can be regarded as points in the two-dimensional coordinate system (x*i* , y*i* ).

First, the mathematical algebraic expression of the circle is:

$$(\mathbf{x} - \mathbf{A})^2 + (\mathbf{y} - \mathbf{B})^2 = R^2 \tag{1}$$

According to (1), another form of the equation of the circle can be obtained:

$$ax^2 + y^2 + ax + by + c = 0\tag{2}$$

That is to say, the parameters of the circle can be obtained by calculating a, b, c. The distance from each point to the center of the circle *di* :

$$\left|\mathbf{d}\right|^2 = \left(\mathbf{x}\_i - \mathbf{A}\right)^2 + \left(\mathbf{y}\_i - \mathbf{B}\right)^2\tag{3}$$

The square difference between the square of the distance from each point to the edge of the circle and the radius is:

$$
\delta\_i = \mathbf{d}\_i^2 - \mathbf{R}\_i^2 \tag{4}
$$

Let Q (a, b, c) be the sum of squares of δ*i* :

Q= *n i*=1 δ*i*  2 (5)

The parameters (a, b, c) are obtained to minimize the value of Q.

Figure 5 shows the binary image of the interface shape of the wind turbine. When the camera has a large angle deflection or the camera cannot collect a complete engine room image, as shown in Fig. 6, this circle fitting algorithm can realize the detection and feedback of the deflection.

**Fig. 5** Cabin docking surface

**Fig. 6** Simulation of large angle deflection

When the camera only captures part of the cabin image, the simple color feature or circular feature algorithm cannot reflect the correct position relationship [8, 9]. However, the circle fitting algorithm based on the least square method can solve this kind of problem to a certain extent.

Firstly, the center coordinates of each bolt hole are obtained by image processing, as shown in Fig. 7.

Then, the circle fitting of the center coordinates of each bolt hole is carried out, and the pixel coordinates of the center of the engine room are obtained, as shown in Fig. 8.

According to the returned information, the position and posture of the hub are adjusted in time to prevent collision accidents and ensure the safety of installation work.

#### **Fig. 8** Fitting the center of the circle

# *3.3 Ranging Sensor Filtering Algorithm Based on Mean Filtering*

The measurement range of the laser ranging sensor used in the experiment is 0.01– 8 m, the measurement blind area is 10 mm, and the measurement error is 2%. When the laser ranging sensor performs distance detection, the detected distance information is unstable and there is a certain error. The software is used to simulate the random error distribution of the ranging sensor within the range, as shown in Fig. 9.

The ranging sensor data is collected at different distances, and several data are randomly collected at the same distance, as shown in Fig. 10, which is the measurement data at a distance of 1138 mm.

Based on the data collected several times, it is concluded that the size of the error of the ranging sensor increases with the increase of the distance when the data measurement is performed. According to the working principle of the range sensor and statistics related theories, the filtering algorithms such as minimum filter, median

**Fig. 9** Random error simulation of ranging sensor

**Fig. 10** Error floating of ranging sensor

filter, mean filter and Kalman filter [10] are used to perform the filtering experiments on the range sensor data in static and dynamic situations respectively, as shown in Fig. 11.

By measuring a number of data at a fixed distance and taking the mean value of a number of data as a reference value. According to the data obtained from many experiments, it can be seen that these filtering methods can well reduce the floating trend of the measurement data. Moreover, the effect of mean filtering is the best.

According to the experimental data under different conditions, when the motion speed is greater than 0.5 m/s, the filtering effect of median filter and Kalman filter is better, and the value is closer to the reference value. When the motion speed is less than 0.2 m/s, the mean filter has the best filtering effect, and the data is stable and close to the reference value.

In addition, the size of the filter also affects the filtering performance. Taking mean filtering as an example, when the motion speed is slow, the larger the filter is, the better the filtering effect is. When the motion speed is fast, the smaller the filter, the better the filtering effect.

**Fig. 11** Comparison of minimum and mean filtering

### *3.4 Coordinate Conversion Based on Image DPI*

In the traditional coordinate transformation method, the internal and external parameters of the camera are calculated by camera calibration, and then the image coordinate system is transformed into the world coordinate system through the camera internal and external parameter matrix and pixel coordinates [11]. When the camera is stationary, this coordinate conversion method can achieve high accuracy and good stability. However, during the installation of the fan hub, the hub and the nacelle are in an unstable state of motion. Therefore, the traditional coordinate transformation method does not fully meet the positioning requirements under this working condition.

Aiming at the inapplicability of traditional coordinate transformation methods, a camera coordinate transformation method based on distance and camera DPI relationship is adopted. At a fixed distance, the number of pixels per inch of the actual distance is fixed. During the movement, the DPI of the camera changes with the distance. By randomly changing the distance between the camera and the measured object within a certain distance range, the relevant data of the camera DPI and distance are obtained.

Then, the curve fitting of camera DPI and distance is carried out by polynomial approximation. Finally, the relationship between camera DPI and distance is obtained, as shown in Fig. 12, Distance in mm.

Through multiple sets of data fitting, the quadratic polynomial equation of camera DPI with distance transformation is obtained.

$$\mathbf{D} = 6.13 \ast 10^{-8} \ast \mathbf{x}^2 + 1.38 \ast 10^{-3} \ast \mathbf{x} - \mathbf{5} \ast 10^{-2} \tag{6}$$

x is the vertical distance between the center of the camera imaging plane and the plane of the cabin, in mm. The measurement for D is mm/pixel, which stands for the

**Fig. 12** Camera DPI and actual distance fitting

reciprocal of the camera's DPI and represents the actual distance that each pixel at a given distance represents.

The derived equation's error squares are equal to 0.02, and the R-square (coefficient of determination) is equal to 0.998.

On the basis of this, it is possible to determine the real distance L of the item that was measured using image processing:

$$\mathbf{L} = \mathbf{D} \ast \mathbf{l} \tag{7}$$

l: The number of pixels in the radius of the image, in pixels.

Taking the engine room center as the origin of the world coordinate system, the actual distance between the engine room center and the camera center is obtained by calculation. Taking the direction perpendicular to the nacelle section as the y-axis, the relative position of the camera relative to the nacelle can be obtained, that is, the position matrix T of the hub, as shown in Fig. 13.

Then the rotation matrix R of the hub is obtained by the gyroscope, and the pose matrix P of the hub can be obtained:

**Fig. 13** Pose coordinate system

$$\mathbf{P} = \mathbf{R} \ast \mathbf{T} \tag{8}$$

R is the initialized rotation matrix. T is the initialized position matrix.

### **4 Experimental Analysis**

Firstly, the error of the visual algorithm itself is evaluated, and the evaluation results can provide reference for the subsequent accuracy improvement. The positioning mark used in the experiment is a blue circular mark with an actual radius of 85 mm, as shown in Fig. 14. The radius of the positioning mark is measured at a fixed distance. The pixel size of the radius of the circle is 47.7 based on the principle of averaging multiple measurements. The visual algorithm is used to process the positioning mark multiple times, and the pixel size of the radius of the circle is measured, and the error of the visual algorithm can be obtained, as shown in Fig. 15.

**Fig. 15** Algorithm error

a) Visual inspection device b) Model of wind turbine nacelle

According to the data obtained from several experiments, the error of the algorithm in detecting the radius is between 0.15 pixels when the conditions are fixed. According to the working conditions of wheel installation, the introduction of mean value filter to optimize the algorithm can keep the error of the algorithm within 0.1 pixels.

On this basis, a ranging sensor is introduced for practical measurement experiments. The positioning markers are subjected to reciprocating motion and arbitrary angle deflection at random speeds within a range of 1 m in front of the camera to simulate the instability of the actual lifting process, as shown in Fig. 16.

The image processing algorithm, ranging sensor filtering algorithm and position detection algorithm shown in the paper are used to carry out the measurement of the radius of the positioning markers for many times, as shown in Fig. 17. The experimental results are compared with the actual radius of the positioning markers to find out the error, and the measured error fluctuation curve is obtained, as shown in Fig. 18.

According to the comprehensive results of multiple sets of experimental data, the position detection method in this paper can realize real-time radius detection of positioning markers in dynamic situations, and the detection error is within 5 mm. The results of multiple experiments show that the position detection method described in this paper has good repeatability. In the process of fan hub installation, the distance between the hub and the nacelle is within 1 m, so the fan hub position detection method described in this paper has reference value for the actual fan hub installation process.

### **5 Conclusion**

In this paper, a method for detecting the position and posture of the fan hub based on machine vision and ranging sensor is proposed. The camera is used to collect the image, and the image is processed twice. The circle fitting algorithm based on the least square method is used to obtain the pixel coordinates of the positioning mark. The relative position relationship between the hub and the cabin is obtained by the relationship between the ranging sensor and the fitted camera DPI with the change of distance. The obtained data is processed and optimized by various filtering algorithms, which improves the accuracy of visual positioning and can meet the positioning requirements under actual working conditions. The feasibility and stability of the wheel pose detection method are verified by experiments, which meets the accuracy requirements of engineering hoisting. The feasibility of applying machine vision to construction machinery is verified.

This visual positioning method can reduce the direct participation of personnel in the process of fan hub hoisting, and return the real-time position information to the crane console and commander. When there is a large angle deflection or abnormal position, the technical personnel can adjust the hoisting process in time to eliminate the risk. It provides a technical reference for the research and development of intelligent hoisting or intelligent crane.

**Acknowledgements** This work is supported by National Natural Science Foundation of China under Grant No. 52275088 and the Fundamental Research Funds for the Central Universities under Grant No. DUT22LAB507.

### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Fault Simulation and Experimental Validation of Accessory Transmission System**

**Yu Sun, Xuyang Cao, Yunbo Yuan, Guang Zhao, Song Ma, and Haofan Li**

**Abstract** As an important part of the aero-engine, the accessory transmission system affects the operation status of the engine, and it is of great significance to carry out fault mechanism analysis and fault identification. This paper takes a certain type of aero-engine accessory transmission system as the research object, flexes the box using ANSYS, establishes a rigid-flexible coupling model under normal working conditions and fault conditions through ADAMS, and studies the vibration characteristics in single fault and multi-fault combination modes, such as unbalance of the shaft system, gear misalignment and gear broken tooth. Considering different load loading conditions, the dynamic simulation of the transmission system is carried out, the shafting displacement and box acceleration response are extracted, the time– frequency domain feature information of the vibration signal is analyzed, and the fault characteristics and fault types are corresponded one-to-one. The test bench for the principle prototype of the accessory transmission system is designed and built, and the experimental research is carried out. The results show that the simulation results have good consistency with the principle prototype test results, which verifies the rationality of the simulation model.

**Keywords** Accessory transmission system · Rigid-flexible coupling · Vibration characteristics · Fault diagnosis

Y. Sun · X. Cao

Y. Yuan (B) · S. Ma School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China e-mail<EMAIL_ADDRESS>
G. Zhao

School of Energy and Power, Dalian University of Technology, Dalian 116024, Liaoning, China

H. Li AVIC Shenyang Aircraft Design and Research Institute, Shenyang 110850, Liaoning, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_31

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

### **1 Introduction**

The accessory transmission system is a key component of the aero-engine, which works in an environment of high-speed, high-temperature and high-load for a long time, and will be subjected to strong impact. Gears, bearings and other components are prone to fatigue damage. Therefore, the study of vibration characteristics under fault conditions is of great practical application significance.

Domestic and foreign scholars have widely used dynamic modeling methods to study the vibration signal characteristics of gear systems. Hong et al. comprehensively considered the time-varying meshing stiffness, meshing damping, and the integrated error of wheel tooth meshing, and simulated the dynamic characteristics of a multi-stage gear transmission system based on ADAMS [1]. Lu et al. established a rigid-flexible coupling model of the gearbox and compared and analyzed the vibration signal characteristics of the gearbox under normal and broken tooth conditions [2]. Loutas et al. established a two-stage gear system model and investigated the effects of the meshing stiffness and tooth flank clearance on the vibration response of the box [3]. Helsen et al. studied the vibration characteristics of the pure torsion model, the rigid six-degree-of-freedom multi-body dynamics model and the rigidflexible coupling model of the planetary gearbox, respectively, and concluded that the rigid-flexible coupling model affects the vibration modes and transfer function of the gearbox [4]. Dąbrowski et al. used ADAMS to establish a rigid-flexible coupling model of planetary gearboxes, and studied the vibration response generated by planetary gear transmission errors and manufacturing errors through the meshing force between gear pairs [5]. Chen et al. quantitatively analyzed the effects by time domain indexes for the rigid-flexible coupling model of planetary gearboxes with different tooth root crack degrees [6]. Wu et al. constructed a fixed-axis rigid-flexible coupling model based on LMS Virtual Lab, and studied the vibration response characteristics of the gearbox under normal, broken tooth and eccentric faults [7].

At present, most of the research focuses on the single-stage gear system and the single gear failure condition of the transmission system vibration response. For the multi-stage transmission system, the coupled fault condition of the box vibration response in research is less, and the lack of test. This paper considers the single-fault and multi-fault combination conditions of the accessory transmission system, such as shafting imblance, gear misalignment and gear broken teeth. The rigid-flexible coupling model of the accessory transmission system is established by ANSYS and ADAMS. The signal characteristics are extracted and compared with the vibration response of the box under normal conditions. Finally, the simulation data is compared with test bench data to verify the rationality of the model.

# **2 Rigid-Flexible Coupling Model of Accessory Transmission System**

Taking a certain type of aero-engine accessory transmission system as the research object, its transmission principle is shown in Fig. 1. Using SOLIDWORKS to establish a three-dimensional model including the gear shaft, spline shaft and box, as shown in Fig. 2, and adding the appropriate coordinate system to determine the relative position of various components of the accessory transmission system.

In order to balance simulation accuracy and model complexity, the model of the accessory transmission system is simplified. Retaining the support hole of the box, ignoring the oil hole, through hole and bolt hole; ignoring the smaller steps and chamfers on the gear shaft; ignoring the key slot on the input shaft and output shaft; and fixing the gearbox body and the gearbox cover as a single unit. According to the actual operation characteristics of the accessory transmission system, respectively, set the revolute joint of the gear shaft and spline shaft relative to the box, the fixed joint of the box and the earth, and the solid–solid contact of the gear pair and the spline pair. The contact collision stiffness coefficient K is related to the shape of the

**Fig. 1** Transmission principle

**Fig. 2** Three-dimensional model

contact surface and the material properties, which are calculated by the formula:

$$\begin{cases} K = \frac{4}{3} \rho^{\frac{1}{2}} E^\*\\ E^\* = \frac{1 - \upsilon\_1^2}{E\_1} + \frac{1 - \upsilon\_2^2}{E\_2} \\ \frac{1}{\rho} = \frac{1}{\rho\_1} + \frac{1}{\rho\_2} \end{cases} \tag{1}$$

where ρ is the comprehensive radius of curvature, *E*∗ is the integrated elasticmodulus, v1 and v2 is the Poisson's ratio of the two meshing gears, *E*1 and *E*2 is the modulus of elasticity of the two meshing gears, and ρ1 and ρ2 is the equivalent radius of curvature of the two meshing gears at the meshing point.

For a pair of spur gears meshing with each other, the radii of curvature of the two tooth profiles at the node are calculated as:

$$\begin{cases} \rho\_1 = \frac{d\_1^{\prime} \sin \alpha^{\prime}}{2} \\ \rho\_2 = \frac{d\_2^{\prime} \sin \alpha^{\prime}}{2} \end{cases} \tag{2}$$

$$d\_1' = d\_1 \frac{\cos \alpha}{\cos \alpha'} \tag{3}$$

where *d* is the diameter of the gear pitch circle, *d* is the diameter of the gear indexing circle, α is the pressure angle of the pitch circle, and α is the pressure angle of the indexing circle.

Considering that the bearings can have a great impact on the system performance and the large number of bearings, the contact model would lead to excessive computation, so it is decided to use the bushing force to simulate the bearings [8]. When using the bushing force to simulate the bearing, the stiffness coefficient *K* and damping coefficient *C* need to be determined. According to the actual situation, the stiffness and the damping in the direction of the rotational degree of freedom are set to be 0, and the stiffness value of the rest of the five degrees of freedom is 1e5 N/mm, and the damping value is 50 Ns/mm.

Compared with the rigid body, the flexible box is more likely to show the vibration characteristics of gear meshing excitation in the simulation process. Therefore, ANSYS is used to flexible the box, which is imported into ADAMS to replace the rigid body structure in the model, and construct the rigid-flexible coupling model of the accessory transmission system, as shown in Figs. 3 and 4.

**Fig. 4** Rigid flexible coupling mode

### **3 Simulation Results and Analysis**

### *3.1 Normal Condition*

The drive is loaded on the revolute joint at the input end of the internal spline shaft, and the load is added on the end of the output shaft in the form of torque. The simulation end time is 0.3 s, and the simulation steps are 25,000. Simulate the normal condition of the accessory transmission system with a rotational speed of 3000 r/min and a load of 20 Nm, extract the displacement of the output shaft and the input shaft, as well as the acceleration of the three measuring points on the box for the time–frequency domain analysis, and position of the measuring points as shown in Fig. 4.

Through simulation, the output shaft speed is 1843.01 r/min, and the error between the theoretical speed and the simulated speed is 0.023% by calculation, theoretical speed as shown in Table 1. The speed of the accessory transmission system under normal conditions is shown in Fig. 5. Analyze the frequency domain of the output shaft speed, as shown in Fig. 6, the larger spectral line of spectrogram amplitude corresponds to the gear meshing frequency ( *f* m32 = 644.82 Hz) and its multiple meshing frequencies, which can verify the feasibility of the model.

According to Fig. 7, it can be seen that the gear meshing vibration response is relatively smooth, and the spectral lines with larger amplitude in the frequency

**Table 1** Operating parameters of the accessory transmission system

**Fig. 7** The time–frequency domain diagram of vibration acceleration

domain correspond to the meshing frequency and its multiple meshing frequencies of the multi-stage gear transmission system. For example, the spectrum components of measuring point 3 are mainly the meshing frequency of 05 gear shaft and 04 gear shaft ( *f* m54 = 996.58 Hz), the double meshing frequencies of 06 gear shaft and 05 gear shaft (2*f* m65 = 2300 Hz), the double meshing frequencies of 03 gear shaft and 02 gear shaft (2*f* m32 = 1289.64 Hz), the triple meshing frequencies (3*f* m32 = 1934.46 Hz), quadruple meshing frequencies (4*f* m32 = 2579.28 Hz).

### *3.2 Unbalance of the Shaft System*

The instability of the transmission system caused by rotor mass imbalance is one of the typical vibration faults of the aero-engine. The test is realized by installing a balancing disk with threaded holes near the gear end of the output shaft to add the amount of unbalance, and the simulation is set up according to the position of the bolts to simulate the unbalance fault of the shaft system with a solid ball with mass. The mass of the counterweight bolt is 0.79 g, the amount of unbalance is 31.6 g mm, and the drive and load are the same as in the normal condition.

**Fig. 8** Time–frequency domain diagram of vibration acceleration at the measuring point 1 and 2

Comparing the output shaft displacement obtained from the normal condition and the unbalanced fault condition as shown in Fig. 10a, the amplitude of the displacement under the fault condition increases in the x and y directions, and the shaft trajectory is elliptical. Extracting the vibration acceleration of the accessory transmission system measuring points 1 and 2 as shown in Fig. 8, compared with the normal condition, the spectral components of the two measuring points are the same as in the normal condition, which are the meshing frequency of the gear drive system and its multiple meshing frequencies.

Due to the small amount of unbalance, the unbalance fault added by the simulation has no obvious effect on the time domain course and spectrogram of the vibration acceleration response of the box. In order to further analyze the fault characteristics, the root mean square (RMS), kurtosis (Ku), waveform factor (WF), crest factor (CF) and impulse factor (IF) of five time-domain indicators are selected to calculate the characteristic differences between the fault condition and the normal condition. The RMS reflects the vibration amplitude, the Ku represents the degree of smoothness of the waveform, and the WF, CF, and IF are used to check whether there is a shock in the signal.

According to the comparison of indicator characteristics at measuring point 1 near the unbalance fault in Fig. 9, when the unbalance fault occurs, other than the RMS, the Ku, WF, CF, and IF of measuring point 1 increase slightly compared with the normal condition.

**Fig. 9** Comparison of time-domain indicators of acceleration signals under unbalanced shafting conditions

**Fig. 10** The displacement comparison results of normal and fault conditions

### *3.3 Gear Misalignment*

In the gear misalignment fault simulation, the fault is simulated by modifying the position and direction of the Marker point where the revolute joint is located. The 05 gear shaft is deflected in the plane formed by its axis and the axis of the 06 gear shaft. The 05 gear shaft near the gear end bearing is deflected by 1 mm in the direction away from the 06 shaft, and the inclination misalignment between the two shafts is 0.57°.

Comparing the input shaft displacement obtained from the normal condition and the misalignment fault condition as shown in Fig. 10b, the shaft trajectory is still elliptical. According to Fig. 11, the time-domain amplitude of measuring point 3 near the 05 fault shaft decreases compared with the normal condition. The spectrum of measuring point 2 generates the meshing frequency ( *f* m65 = 1150 Hz) and triple meshing frequencies (3*f* m65 =3450 Hz) of 06 gear shaft and 05 faulty gear shaft on the basis of the spectrum component of the normal condition. The spectrum of measuring point 3 generates the 06 gear shaft and 05 faulty gear shaft triple meshing frequencies (3*f* m65 = 3450 Hz) and the triple meshing frequencies (3*f* m54 = 2989.74 Hz) of the 05 faulty gear shaft with the 04 gear shaft on the basis of the spectral components of the normal condition.

According to the comparison of indicator characteristics at measuring point 3 near the misalignment fault in Fig. 12, when the misalignment fault occurs, the RMS, Ku, CF, and IF decrease slightly compared with the normal condition.

**Fig. 11** Time–frequency domain diagram of vibration acceleration at the measuring point 2 and 3

**Fig. 13** Time–frequency domain diagram of acceleration at measuring points 1 and 2

### *3.4 Gear Broken Tooth*

Considering the characteristics of processing and assembly, a broken tooth fault is manually set up in the 03 gear shaft, the normal gear in the model is replaced by a broken tooth at the root of one of its gear teeth, and the rigid-flexible coupling model of the accessory transmission system with a broken tooth fault is constructed.

Comparing the output shaft displacement obtained from the normal condition and the gear broken tooth fault condition as shown in Fig. 10c, the amplitude of the displacement under the fault condition increases in the x and y directions, and the shaft trajectory is elliptical. According to Fig. 13, compared with the normal condition of the box vibration time-domain, the broken teeth fault condition of the vibration acceleration has a significant impact, and the impact signal interval time is 0.048 s, which corresponds to the reciprocal of the rotation frequency ( *f* z3 = 20.8 Hz) of the 03 fault gear shaft. The spectral components of the fault condition are similar to those of the normal condition, and the amplitude corresponding to the meshing frequency and its multiple meshing frequency spectral lines increases. From the refined spectrum near the meshing frequency (3*f* m32 = 1934.46 Hz), it can be seen that there exists a faulty shaft rotation frequency ( *f* z3 = 20.8 Hz) and its frequency doubling modulation at the meshing frequency. The modulation sidebands are asymmetrically distributed.

As can be seen from Fig. 14, the values of the indicators at measuring points 1 and 2 close to the fault increase.

### *3.5 Unbalance of the Shaft System and Gear Broken Tooth*

When implementing the coupled fault, refer to the single fault implementation mode. A coupled fault rigid-flexible model is constructed by adding a broken 03 gear shaft tooth fault to the accessory transmission system, while a sphere with mass is added to the 02 gear shaft end.

**Fig. 14** Comparison of time-domain indicators of acceleration signals at measuring points 1 and 2

Comparison of normal conditions and coupled fault conditions of the output and input shaft displacement, as shown in Fig. 15, shows that coupled fault conditions of the input and output shaft displacement increase the amplitude in the x and y directions, and the shaft trajectory is elliptical. As can be seen from Fig. 16, due to the small amount of unbalance, the shaft system imbalance and gear broken teeth coupling fault time–frequency domain analysis more intuitively show the characteristics of a broken teeth fault. In order to further explore its fault characteristics, calculate the time-domain indicators. According to Fig. 17, the five indicator values of measuring points 1 and 2 have increased. Compared with the single gear broken fault, the coupling fault adds the unbalance mass, and the increase at measuring point 1 near the shaft system imbalance fault is slightly reduced.

**Fig. 15** Output shaft, input shaft displacement

**Fig. 16** Time–frequency domain diagram of vibration acceleration at measuring points 1 and 2

**Fig. 17** Comparison of time-domain indicators of acceleration signals under coupled fault conditions

### **4 Experimental Verification**

In order to verify the simulation results, the test bench for the accessory transmission system is constructed as shown in Fig. 18. The arrangement of the acceleration sensor on the box is consistent with the position of the measuring point of the simulation

**Fig. 18** Test bench sensor installation position

model, and the eddy current displacement sensor is set at the end of the output and input shafts, respectively. The loading conditions are the same as in the simulation.

From Fig. 19, it can be seen that the axial trajectory is elliptical, and the amplitude in the x and y directions increases slightly under broken tooth and coupling faults. According to Fig. 20, under normal conditions, the time-domain simulation results of the vibration acceleration of the three measuring points of the box are consistent with the main test results, the main components in the spectrum are the gear shaft meshing frequency and multiple meshing frequencies. Under the fault condition, extracting the acceleration of the measuring points close to the fault shaft for comparison. The gear broken fault and coupling fault in the simulation results of the impact of measuring point 2 is not obvious. The sidebands appear on both sides of the spectrum meshing frequency of the test results, and the comparison results of the remaining measuring points meet the consistency of simulation and test.

**Fig. 19** Displacement test results

**Fig. 20** Vibration acceleration simulation and test results at measuring points

# **5 Conclusion**

In this paper, by constructing the rigid-flexible coupling model of the accessory transmission system, the time–frequency domain characteristics under fault excitation are studied. The conclusions are as follows:

**Acknowledgements** This work is supported by the National Natural Science Foundation of China (No. 52275088) and the Fundamental Research Funds for the Central Universities (No. DUT22LAB507).

### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Adaptive Control of Drilling Rig Power Head Speed**

**Xinxin Xu and Wanzhi Cai**

**Abstract** In the drilling process, the adjustment of drill pipe rotational speed is essential to accommodate the variations encountered in different geological strata. However, the existing low automation level in drilling rigs and heavy operator reliance often result in suboptimal speed regulation, leading to low drilling efficiency. An adaptive control approach for power head speed based on a fuzzy clustering algorithm was proposed, which utilizes the fuzzy clustering algorithm (FCM) to cluster drilling parameters, including drilling pressure, rotational speed, and drilling speed, to identify of distinct strata. Furthermore, the rotational speeds of the clustering centers of different strata were used as the input rotational speeds of its control system, the simulation results show that the speed control can be realized by controlling the opening of the valve port, and the fuzzy PID control is used to realize the adaptive control of the speed and the tracking of the speed curve.

**Keywords** Stratigraphic recognition · Adaptive control · Clustering algorithm · Fuzzy PID · Power head

# **1 Introduction**

In the drilling process, due to the different strata encountered, it is necessary to adjust the rotational speed of the drill pipe to adapt to different formations. The automation degree of the drilling rig is not high, and the drilling effect is highly dependent on the operator. When the encountered formation changes are complicated, owing to the

X. Xu (B) · W. Cai

#### X. Xu

Key Laboratory of Intelligent Manufacturing of Construction Machinery, Anhui 230009, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_32

National Engineering Research Center for Highway Maintenance Equipment, Chang'an University, Xi'an 710064, China e-mail<EMAIL_ADDRESS>
Henan Key Laboratory of High. Grade Highway Detection and Maintenance Technology, Xinxiang 453003, China

limitations of the operator's experience and the influence of the change of the formation environment; it is often difficult for the operator's reaction speed to adapt to the change of the working conditions. The speed regulation process may have hysteresis, and the speed regulation effect is not ideal, resulting in poor drilling efficiency [1]. The proximity of the operating table to the orifice is a safety hazard. At present, the control of drilling parameters is still in the state of theoretical research, mainly due to the complexity of the drilling process mechanism, the correlation among the variables, is the nonlinear and uncertain process. The interference factors are many random and difficult to accurately measure and there is no clear functional relationship between the monitoring parameters and the drilling parameters. However, it is impossible to describe the drilling process with a precise mathematical model. In order to improve the adaptive ability of drilling rigs in complex rock formations, Xie et al. [1], used the ratio gong method to identify the formation, which provided conditions for realizing the adaptive control of rotational speed. Xu et al. [2], analyzed the drilling parameters collected by a real-time monitoring system and showed that it is effective to judge the formation based on the follow-drilling parameters. Wu et al. [3], established a propulsion adaptive control system, which improved the adaptive ability of the drilling rig in complex formations and controlled the degree of drilling deflection. Cheng et al. [4], proposed a synergistic adaptive control method between the propulsion system and the rotary system, and the optimal rotational speed was determined by using the collected parameters of the drilling with the drill. Lu et al. [5], designed a fuzzy PID control system to realize the stable control of drilling pressure of drilling rig. Chang et al. [6], used BP neural network technology to analyze historical drilling data and realize adaptive control of drilling pressure. H. Graham investigated a system identification method using ROP and RPM as inputs and WOB and Torque as outputs for identification. The method can be applied in real time during the drilling process to recognize typical drilling dynamics for adaptive control [7].

Machine learning algorithm is the research focus of lithology identification, which has the advantages of high precision, intelligence, high efficiency, accuracy and realtime acquisition of stratigraphic information compared with the traditional method of lithology identification by monitoring the follow-drill parameters. Yue et al. [8], summarized the current status of the application of various types of commonly used machine learning algorithms in the field of lithology identification, and compared the advantages and disadvantages of various types of algorithms in the application of lithology identification. Gao et al. [9], established a K-means based lithology classification model to realize the classification of lithology. Zhang et al. [10], proposed a clustering algorithm to classify rock types on core data to realize the classification and identification of lithology. Zhang et al. [11]. established a group clustering algorithm to recognize volcanic lithology and compared it with K-means and other algorithms, and the results showed that this method is a more effective lithology recognition algorithm with high accuracy.

L. A. Castañeda et al. used adaptive control for solving problems with uncertain dynamic models. The results show that the adaptive controller also outperforms the PID controller [12]. In this paper, the fuzzy clustering algorithm (FCM) technology is introduced into the control system of power head speed to realize the discrimination of strata, and then realized the adaptive control of speed, Laying the theoretical foundation for automation and intelligence of drilling rigs. AMESim and Simulink are used for joint simulation to realize closed-loop speed regulation and speed curve tracking by fuzzy PID algorithm.

# **2 Modeling and Analysis of the Powerhead Hydraulic System**

The power head rotary circuit consists of Rexroth's A8VO variable double pumps, multi-way valves, and variable motors. The main pump A8VO double pump supplies oil to the LUDV valve. The LUDV system can be divided into three parts, including the throttling part, the pressure compensation part and the reversing part, in which the throttling part regulates the flow size of the system by controlling the size of the throttling port area; the pressure compensation part maintains the stability of the pressure difference between the front and rear of the valve; and the reversing part controls the flow direction of the oil to control the motor's forward and reverse rotation. Due to the complexity of the power head system, in order to simplify the system model to speed up the simulation, the main pump is represented by a variable pump, and the main component parameters are shown in Table 1.

The modeling of the hydraulic system of the power head is shown in Fig. 1, because the different loads faced by the power head in different strata, the rotational speed of the motor of the power head in different strata (sand and gravel layer, coal layer, and rock layer) is analyzed through the simulation of AMESim software, and the results of simulation are shown in Fig. 2. From 0 to10 s, the higher motor speed is 1414.4 r/min for load of 168.8 Nm. From 10 to 30 s, the motor speed is 675.25 r/ min for load of 976.7 Nm. From 30 to 40 s, the motor speed is 466.2 r/min for load of 1012.8 Nm. The results show that the rotational speeds of the motor of the motor in different strata are different, with the lowest rotational speed of the sand and gravel layer, the rotational speed of the coal layer is higher than the one of the sand and gravel layer, and the rotational speed of the rock layer is the highest.

**Fig. 1** Hydraulic system model

### **3 Stratigraphic Identification Methods**

In the drilling process, the different mechanical properties possessed by different strata, and the drilling parameters can obviously reflect their differences [13, 14]. The drilling parameters such as drilling pressure, rotational speed and drilling speed are selected as the characteristic parameters. In this paper, the relevant drilling parameters are obtained through indoor drilling experiment mainly based on the variability of rotational speed in different formations. The drilling test rig consists of drilling rig, hydraulic pumping station, water circulation system and other parts, and the hydraulic system of the drilling rig consists of relief valves, hydraulic cylinders, oil tanks and other components. The relief valve controls the pressure of the whole hydraulic system. The drilling rig test bench is shown in Fig. 3, and the drilling test bench is utilized for the acquisition of parameters with drilling, including rotational speed, drilling speed, and drilling pressure.

In order to adapt the power head of the drilling rig to different strata during the drilling process, it is necessary to manually adjust the rotational speed in the past, disadvantages of speed lag and low efficiency. Clustering is an unsupervised machine learning method, the goal is to collect and categorize data, and the clustering algorithm can collect the drilling parameters to analyze and judge the stratum. In order to select the appropriate rotational speed, it overcomes the disadvantage of manually adjusting the speed. In the clustering process, data objects similar to each other will be divided into the same subset, and the whole observation data will be divided into several subsets. Therefore, FCM algorithm is used as the algorithm for strata identification [15].

A number of problems in different areas have been effectively solved through the use of FCM and its different variants.

Fuzzy C-mean clustering algorithm (FCM) is to classify *n* vectors into *c* classes, the clustering center of is *V*{*v*1,*v*2…*v*n}, this is accomplished by finding the minima of the objective function of Eq. (1).

$$J(U,V) = \sum\_{i=1}^{c} J\_i u\_{ij}^m = \sum\_{i=1}^{c} \sum\_{j=1}^{n} u\_{ij}^m d\_{ij}^2 \tag{1}$$

Affiliations satisfy the condition,

**Fig. 3** Drilling test rig

PDC drill bit Drill pipe Slewer Racks

$$\sum\_{i=1}^{c} u\_{ij} = 1, 0 \le u\_{ij} \le 1, i = 1, 2, \dots, n \tag{2}$$

The algorithm obtains the clustering centers by minimizing the objective function, it is solved by the Lagrange multiplier method so that its minimum value satisfies the condition:

$$V\_i = \frac{\sum\_{j=1}^{n} u\_{ij}^m \mathbf{x}\_i}{\sum\_{j}^{n} u\_{ij}^m} \tag{3}$$

$$U\_{ij} = \frac{1}{\sum\_{k=1}^{c} (\frac{d\_{ij}}{d\_{kj}})^{\frac{1}{m-1}}} \tag{4}$$

FCM algorithm steps as follows,

The clustering results of the drilling parameters based on the FCM algorithm are shown in the Fig. 4, the results show that the typical drilling parameters of rotational speed, drilling speed and drilling pressure are different for different formations, with obvious partitions, it can be concluded that the application of the clustering algorithm to the identification of formations in the drilling process is reliable and has obvious effects. The cluster center obtained by the clustering algorithm is the optimal drilling parameter for a certain formation.

In the drilling process, through the clustering algorithm of the drilling parameters collected by the sensor (rotational speed, drilling pressure, drilling speed, etc.) to analyze and process the information of the current stratum, and then select the optimal rotational speed for the realization of adaptive control of rotational speed to provide the conditions.

# **4 Fuzzy Adaptive PID Control Strategy for Power Head Speed**

When facing the nonlinear system and complex working conditions with variable parameters, PID control is less effective. Fuzzy control can control the nonlinear system without obtaining an accurate mathematical model, but the control accuracy is not satisfactory. The use of fuzzy adaptive PID control integrates the advantages of

**Fig. 4** Clustering results

the two, and achieves the purpose of online adaptive adjustment of PID parameters, enhanced the control performance. The principle of fuzzy adaptive PID control is to take the error and the rate of change of the error as the input, through the rules obtained from expert experience, to establish the proportional, differential and integral three coefficients of the fuzzy control table for the control of the system [16], and its principle is shown in Fig. 5.

In this paper, a typical two-dimensional fuzzy controller is used, the inputs are motor speed deviation e and the deviation change rate ec, and the three output parameters *Kp*, *Ki* and *Kd* are obtained through four processes: quantization, fuzzification, fuzzy inference and defuzzification. Set the fuzzy domain of input deviation *e* = [−3,3], deviation change rate *ec* = [−3,3], the fuzzy domain of output *Kp* = [−0.3,0.3], the fuzzy domain of *Ki* = [−0.06,0.06], and the fuzzy domain of *Kd*  = [−3,3]. The input and output quantities are divided into seven fuzzy subsets {*NB*, *NM*, *NS*, *ZO*, *PS*, *PM*, *PB*}.

**Fig. 5** Fuzzy adaptive controller principle

# **5 AMESim and Simulink Co-simulation**

### *5.1 Control Modeling*

Simulink is a modular graphical environment, a visual simulation tool used for model building and simulation analysis of results in many fields, with the advantages of clear structure and high efficiency. Moreover, MATLAB algorithms can be integrated into the model in Simulink, and the simulation results can be exported to MATLAB for further research and analysis [17].

In this paper, visual, MATLAB and AMESim are used. A joint simulation model is created in AMESim for simulation to generate S-functions [18]. In Simulink environment, the S-function is called and the simulation model is created. The control system built in Simulink is shown in Fig. 6.

**Fig. 6** Simulink model

**Fig. 7** Three stratigraphic rotational speed responses

### *5.2 Response Under Three Kinds of Rotational Speeds*

Taking the rotational speeds of three different strata as input signals, by adjusting the three parameters of the fuzzy PID, the system can achieve a better control effect when *KP* = 0.1, *Ki* = 1, and *Kd* = 0. When the simulation time is 40 s, the response time is 2 s, the simulation results shown in Fig. 7.

### *5.3 Speed Curve Tracking Characteristics*

The drilling characteristics of different strata are different, in order to improve the drilling efficiency, it is necessary to make the dynamic response characteristics of the power head system is good, when the stratum change is the power head can realtime speed adjustment to adapt to different formations. The power head system uses sinusoidal signal and square wave signal as input signals to simulate the rotational speed change of different formations to observe the following effect of the power head, and it can be seen that the dynamic tracking effect of the speed is better through the adjustment of the three parameters of fuzzy PID. Figure 8 shows the opening of the valve port under the sinusoidal signal, and Fig. 9 shows the pressure change. Figure10 shows the following curves of power head speed under sinusoidal and square wave signals. Figure 11 shows the opening of the valve port under the square wave signal, and Fig. 12 shows the pressure change.

**Fig. 9** Motor inlet pressure with sinusoidal signal

# **6 Conclusions**

**Fig. 10** Power head speed tracking response

**Fig. 12** Motor inlet pressure under square wave signal

dynamic response of the system was simulated and analyzed with sinusoidal and square wave signals as input signals, and the opening of the valve port and the change of pressure under sinusoidal signals were also analyzed.

(3) Under the same PID parameters, the powerhead speed tracks the input signal better, both in the sine signal and in the square wave model. A smaller amount of overshooting, about 3.08%, was present only in the case of signaling mutations. The power head speed achieves a good speed curve tracking effect.

**Acknowledgements** The authors would like to thank Shaanxi Provincial Science and Technology Department (No. 2022PT-30), Key Technological Special Project of Xinxiang city (No. 22ZD013) and Key Laboratory of Intelligent Manufacturing of Construction Machinery (No. IMCM2021KF02).

### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Method of Assembling Clearance Measuring of Oil Film Bearing in Tandem Cold Mill**

**Bo Sun, Chuanjian Liu, Weimin Zhou, Zhan Xu, and Heng Cao**

**Abstract** In order to reduce the certain number of oil film bearing burning accident, this paper proposes a novel method used to test static pressure pipes and measure the assembling clearance between sleeve and bush inside the bearing. On account of the theory of lubrication and the working mechanism for oil film bearing, through the development of a new set of hydrostatic oil supply system and test platform, provide the static supporting pressure and the disassembling and locking pressure. And based on this method, the assembling clearance measuring was carried out. The results show that the proposed measuring method has the ability to support the disassembling and assembling process of the oil film bearing. In addition, this method provides an important basis for the engineers of cold rolling mill on site to adjust the assembling clearance of oil film bearing. The application of this method effectively reduces the online burning accidents of the oil film bearing.

**Keywords** Cold rolling · Oil film bearing · Sleeve and bush · Assembling clearance

# **1 Introduction**

Cold rolling is a form of cold working process performed below the recrystallization temperature. Undergoing multi-stage processing including pickling, cold rolling, heat treatment and coating, hot-rolled strip eventually transforms into cold-rolled products [1]. The majority of modern large-scale tandem cold mills characterized by

© The Author(s) 2024

B. Sun (B) · C. Liu

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail<EMAIL_ADDRESS>
W. Zhou · Z. Xu Baoshan Iron and Steel Co., Ltd., Shanghai 201900, China

H. Cao

School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_33

large size, heavy load, continuous operation and automation, use oil film bearing as the heart of whole machine [2]. With the function to provide stable loading capacity for rolling process, oil film bearing has the feature of high working load, high operating speed, and harsh working environments [3]. If the running stability of the oil film bearing is compromised, their periodic errors would be beyond the compensation capability of various automatic control systems and directly impact the quality of the finished sheet. Furthermore, in case of an online burning incident in the oil film bearing, the entire mill would be forced to shut down, resulting in significant economic losses [4].

Consider a hybrid bearing with a design of double static pressure chambers for oil film bearing. The failure of one of the chambers will cause the clearance between sleeve and bush to tilt, accelerating the wear of Babbitt alloy [5], and indirectly reducing the service life of the bearing [6]. Therefore, ensuring the state of the static pressure chambers is particularly important. To ensure no leakage in the static pressure pipe system during each assembling process, a roller grinding workshop used to adopt a kind of conical tool recommended by Morgan Company to mutually support the upper and lower hydrostatic oil inlet, as a sealing measure. This method was effective when initially used, but after multiple uses, it caused certain damage to the inlet and resulted in the bump of nearby Babbitt alloy. The bump sealed off the inlet, causing severe negative impact. Morgan also recommends an online test method of the static pressure chamber using an online hydrostatic pump. However, this method occupies production time and is not accurate due to the mutual influence of the upper and lower rollers.

Besides, the clearance is a principal assembling parameter of the bearing. It plays a crucial role in load carrying, temperature rise and other factors during the operation of the bearing [7, 8]. So far, only the outer diameter of the sleeve and the inner diameter of the bush are measured in the workshop before assembling, while the actual assembling clearance after assembling is unable to measure. When the clearance is too small, the heat generated from the friction between the sleeve and bush accumulates continuously, causing sudden locking of the high-speed rotating roller immediately, thus causing burning accidents [9]. As the internal sleeves and bushes gradually age, most of the bushes in the workshop have undergone multiple repairs, leading to an increasing trend in the clearances. According to recent records for several years, the average number of sleeve damages per year is 19, and this number even skyrocketed to 28 in the last recorded year. What is particularly serious is that 8 of them were online damages, which bring out the shut-down of the machines to replace the support rollers. The total downtime reached 22 h during which the direct economic loss from sleeve repairs amounted to 2.05 million yuan. Therefore, it is necessary to propose a novel method for the static pressure pipe system test and the assembling clearance measuring, to avoid leakage in the system and master the corresponding relationship between actual assembling clearance and on-site use status.

### **2 Theory of Lubrication for Oil Film Bearing**

Bearings are essential components in mechanical systems. Based on the hydrodynamic principles, once the shaft starts rotating, it continuously squeezes lubricating oil with a certain viscosity into the converging gap. As the shaft sucks the lubricating oil into the wedge-shaped gap, the compression of the oil generates distributed resistance, which is termed the hydrodynamic pressure [10]. When the sum of the hydrodynamic pressures in the load direction is able to balance the external load, a complete layer of pressure oil film is formed between the shaft and the bearing. The oil film causes the surfaces of the shaft and the bearing to separate from each other, therefore resulting in pure liquid friction. The whole process above is known as the hydrodynamic lubrication mechanism [11].

The fluid lubrication film that exists between Surface 1 and Surface 2, and the velocity of two surfaces are shown in Fig. 1. The Reynolds equation can be derived as follows from the velocity components *U*1, *V*1 and *W*1 on surface 1, and the velocity components *U*2, *V*2 and *W*2 on surface 2 [12]:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{y}} \right) = 12 \frac{\partial (\overline{U}h)}{\partial \mathbf{x}} + 12 \frac{\partial h}{\partial t} \tag{1}$$

Here, *U* = 1 2(*U*1 + *U*2), *h* = *h*2 − *h*1, η represents the viscosity of the lubricating oil.

For oil film bearing, it is usually a fixed bearing and a rotating journal, implying that *U*1 = 0 and *U*2 =*U*. Considering the steady state situation, the Reynolds equation has the form which is commonly used in engineering:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{y}} \right) = 6U \frac{\partial h}{\partial \mathbf{x}} \tag{2}$$

Meanwhile, introduce the Reynolds boundary condition, assuming that the breakthrough of the oil film occurs at a divergent gap downstream from the minimum oil film thickness. At this point, the pressure and pressure gradient are both zero. So, the condition for complete termination boundary of the oil film is defined by:

$$p = p\_a, \quad \text{and} \quad \frac{\partial p}{\partial \alpha} = 0 \tag{3}$$

When adopting hybrid journal bearing as oil film bearing, the hybrid journal bearing is supplied with oil through hydrostatic system during the startup, braking or low-speed operation of the bearing. The hydrostatic oil film separates the Babbitt alloy layer on the surface of sleeve and bush, achieving the effect of hydrostatic levitation to improve performance of the bearing under low-speed operation [13]. As the rotational speed of the bearing gradually increases, the bearing reaches a hydrostatic-hydrodynamic hybrid lubrication state due to the oil film formation. During the operation of the bearing, static pressure is mainly utilized, while dynamic pressure act as a supplement to compensate for static pressure loss with hydrodynamic effect [14].

### **3 Test Method**

The test method mainly consists of four parts, (1) static clearance test stand for oil film bearing, (2) electric-hydraulic control loading trolley, (3) loading trolley and (4) assembling clearance measuring module. Considering the complexity of the work environment in industrial sites, it is necessary to pre-layout the equipment involved in the testing method. In accordance with the dimensions obtained from surveying and mapping, a three-dimensional solid model of the work area is established as Fig. 2.

Figure 3 shows the process of assembling clearance measuring. The oil film bearing is placed on a bracket after its complete assembly. Then, communicate the hydrostatic outlet of the electric-hydraulic control loading trolley with the hydrostatic inlet on the bearing seat through a hydraulic hose. The lubricating oil is injected by means of hydrostatic oil supply, to prevent the sleeve and bush from moving up and down, which would cause collisions and wear on roller. Simultaneously, push the loading trolley below the bearing seat, and output control commands through test stand to make the piston rod on the loading trolley move up and down, thereby driving the bearing seat to move similarly. When the roller is stationary, the displacement of the bearing seat from its natural resting position to the highest position shall be the target assembly clearance. The clearance value is calculated using a digital dial gauge installed on the bearing seat, which has remote transmission capability to measure the clearance.

# *3.1 Static Clearance Test Stand for Oil Film Bearing*

As an operation terminal, the test stand is the primary human–machine interface device for performing test tasks and can collect test data in real-time. It is equipped with a computer analysis module inside for processing data, as well as a liquid crystal display with digital voltmeter display module on the front panel. The buttons on the control panel are responsible for process control throughout the entire test process. In addition, the test stand has a PLC control module and possesses data printing functionality.

### *3.2 Electric-Hydraulic Control Loading Trolley*

The electric-hydraulic control loading trolley is a movable hydrostatic oil supply system that provides test pressure for various test conditions. According to the test requirements, the trolley needs to fulfill the following functions: be able to provide a pressure of 8 MPa for locking the oil film bearing; provide a pressure of 10 MPa for disassembling the bearing; and provide a pressure for static pressure test of the bearing, with a theoretically designed value of up to 80 MPa.

The principle diagram of the hydraulic system for the trolley is shown in Fig. 4. The maximum working pressure of the system reaches 80 MPa. The system uses a highpressure ball valve to switch pressure between ultra-high and high, with the secondary pressure being controlled by a proportional relief valve to avoid the impact of primary pressure. The primary pressure is directly provided by a piston pump for the static pressure test. In terms of unloading, use the median function of electromagnetic directional valves for the process, with overflow valves serving as safety valves.

The whole specific test process can be subdivided into 9 conditions with corresponding action sequence of each electromagnet under different states, as shown in Table 1. When pump 1–1 starts, electromagnet 7YA is energized and the system starts running in low pressure mode. The system remains in an unloading condition until any operation is performed. At this time, all electromagnets except for 7YA and 8YA are in a power-off state and all electromagnetic valves are in the median function state. Currently with the system being in condition 7, the system pressure can be adjusted by proportional relief valve 1–4. The states of all electromagnets in condition 2, 4 and 6 are same as those in condition 7.

When the system is loading for locking or disassembling, with three electromagnets adjusting the pressure, test oil passes through electromagnetic directional valves

**Fig. 4** The hydraulic principle diagram of the hydrostatic oil supply system

**Table 1** Sequence table for the actions of electromagnets

1–2 and 1–5. Then, a power-on electromagnet 6YA would introduce the oil into the rodless chamber of locking cylinder 1–9, leading to condition 1; or a power-on electromagnet 5YA would introduce the oil into the rod side, leading to condition 3. In condition 5, three energized electromagnets adjust system pressure for static pressure test, ensuring that the test oil enters static pressure chamber 1–10. In condition 8, the system pressure is adjusted by a high-low pressure system consisting of overflow valves. The operations in condition 9 is similar to those in condition 5. When the oil pressure lifts the bearing seat, the head of the digital dial gauge moves accordingly, thus measuring the clearance.

### *3.3 Loading Trolley*

If the loading pressure of the oil pressure control module is inadequate to lift the bearing seat, it is necessary to assist with the loading pressure output by the auxiliary oil pressure control module on the loading trolley. Owing to the load-bearing surface of the oil film bearing being on the underside, the force applied to the bearing seat by the loading trolley needs to be upward. The difference between the applied force and the self-weight of the bearing seat is balanced by the oil film pressure, which is supplied by hydrostatic system to avoid rigid contact between sleeve and bush.

### **4 Analysis of Test Result**

This testing method has been conducted on-site at a cold rolling plant, as the photos in Fig. 5 show. Table 2 presents the relationship between the assembling clearance and burning accident. The following ten assembling clearance test values are listed, and the tracking records of whether there were burning accidents during the operation of the oil film bearing under these assembling clearances are provided.

For oil film bearings, the initial design value of assembling clearance is generally taken between 0.3 and 2.0‰ of the nominal diameter. The permissible range for the assembling clearance is between 0.2355 and 1.57 mm, calculated based on a 785 mm outer diameter of the sleeve. However, it needs to be noted that this range is too broad and vague so that it can only serve as a reference.

Figure 6 shows the actual data on assembling clearance collected using an offline test device for oil film bearing. There are a total of 33 assembling data points, with each point representing a clearance value. The triangles in Fig. 6 represent the clearance values where burning accidents occurred, while the dots represent the values without accidents. Taking 1.1 mm as the critical threshold, it is evident that all assembling clearance values resulting in burning accidents are below 1.1 mm. The 7 triangles in 21 data points indicate a staggering probability of 33.33% for burning accidents. In contrast, there have been no burning accident above 1.1 mm. Therefore, after analyzing with on-site engineers, it can be concluded that the danger zone for assembling clearance is below 1.1 mm, while the safe zone is above 1.1 mm. It is suggested to set the assembling clearance for oil film bearing starting from 1.1 mm,

**Fig. 5** The industrial site test photos

**Table 2** Table for the measurement results of the clearance between sleeve and bush

**Fig. 6** The relationship between the assembling clearance and burning accident

with an upper limit not exceeding the maximum value 1.4 mm in the diagram. If the safety factor is set to a larger value, it can be between 1.2 and 1.4 mm, but must not exceed 1.57 mm.

### **5 Conclusion**

Here are a few suggestions for selecting assembling clearance between sleeve and bush:

In conclusion, it is a feasible solution to increase the minimum value of the assembling clearance from below 1.1 mm to above the safety threshold of 1.1 mm without exceeding the maximum clearance value, which can prevent the clearance from being in the danger zone. The clearance can be effectively controlled by the measuring device for the clearance between sleeve and bush.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.