Effect of Roller Angle on Formability in Rotary Forming of Spiral Corrugated Tubes

Abstract

Spiral corrugated tubes are widely utilized to enhance the performance of heat exchangers. However, they are typically formed via hydroforming, which renders efficient manufacturing challenging. Therefore, this study presents a rotary forming method using multiple rollers for the continuous production of spiral corrugated tubes. During the forming process, the rollers rotate around the tube, pressing against its outer surface, and the tube moves axially, forming spiral grooves. This study experimentally evaluated the effects of varying roller angles on formability by analyzing maximum rotation speed, outside diameter, thickness distribution, groove depth, and peak pitch. The experiments were performed thrice under each condition to ensure reproducibility. The results indicate that the formable rotation speed increases by 40% when the roller angle is adjusted from 32° to 40°. For the same rotational speed, a larger roller angle prevents stress concentration. As the roller angle decreases, the outside diameter also decreases, and the groove depth and peak pitch tend to increase. Under a roller angle of 40° and a rotational speed of 150 rpm, the thickness deviation ratio of the formed product is only 0.13, demonstrating improved uniformity.

1. Introduction

Heat exchangers are indispensable in various fields, such as industrial machinery, automobiles, nuclear power, and the space industry. In recent years, the push for a sustainable society has heightened the demand for high-performance heat exchangers. An effective way to enhance their performance is by utilizing high-performance heat transfer tubes. Among various heat transfer tubes, corrugated tubes are widely employed owing to their excellent heat transfer capabilities [1]. In particular, spiral corrugated tubes exhibit 2.4–3.7 times the heat transfer performance of conventional smooth tubes [2]. Additionally, several researchers have reported that the performance of spiral corrugated tubes can be further improved by optimizing their geometry [3,4]. Therefore, the demand for spiral corrugated tubes has surged, necessitating the development of efficient methods for manufacturing tubes with varied geometries.

Corrugated tubes are typically manufactured using the hydroforming method (Figure 1). Safari et al. [5] investigated the effects of die stroke and die geometry, which are critical factors in hydroforming, on product characteristics. The authors found that excessive thickness reduction in the tube wall is the primary cause of failure. However, using a die with a large corner radius can mitigate this issue and enhance formability. Jiang et al. [6] examined the effect of internal pressure on wall-thickness reduction during hydroforming. The results revealed that excessive internal pressure causes thinning of the tube wall, negatively impacting formability. Conversely, insufficient internal pressure leads to clearance between the tube and die. Therefore, to achieve high shape accuracy, internal pressure must be maintained within an appropriate range. Yuan et al. [7] analyzed the deformation behavior of high-pressure corrugated tubes using numerical analysis. The authors observed that the thickness distribution followed a linear pattern from the peak to the groove of the wave, with a maximum thickness reduction ratio of up to 20%.

Spiral corrugated tubes are also primarily fabricated via hydroforming. Shi et al. [8] successfully produced a high-aspect-ratio spiral corrugated tube at a temperature and an internal pressure of 700 °C and 70 MPa, respectively. In conventional hydroforming, pressure is applied inside a tube using oil or gas. Shi et al. [9] proposed an alternate method for manufacturing spiral corrugated tubes using mandrels and an external high-pressure to avoid fractures caused by thickness reduction. Their method demonstrated high shape accuracy and effectiveness in manufacturing spiral corrugated tubes. However, hydroforming requires expensive equipment such as high sealing components. Additionally, the complex internal pressurization process should be precisely controlled during forming [10]. Therefore, novel forming methods have been developed to replace hydroforming. Jin et al. [11] introduced an incremental forming method that utilizes a specially shaped die to improve forming efficiency. This method imposes no restrictions on the length of the parent tube, making it suitable for diverse industrial applications. Additionally, it can be performed using conventional hydraulic or mechanical presses, offering a considerable advantage in terms of equipment costs. However, manufacturing products with different shapes remains costly, as each target shape requires a custom-made forging die. Grzancic et al. [12] proposed a versatile incremental profile forming technique. This method enables the fabrication of tubes with different cross-sectional shapes along the longitudinal axis of the tube. However, despite extensive research, the challenge of simultaneously achieving high cost-effectiveness and production speed remains unresolved.

One solution to improve the efficiency of corrugated tube manufacturing is rotational forming. It offers high forming speed and requires only simple and inexpensive forming equipment. Moreover, it eliminates the need to fabricate different dies for different target shapes. Therefore, the rotational forming method is highly cost-effective and suitable for the large-scale production of corrugated tubes. Kuss and Buchmayr [13] presented a theoretical formula for the forming load in tube flaring with planetary balls and mandrels. Kuss and Buchmayr [13] validated their findings via experiments and numerical analyses. Hirama et al. [14] designed a planetary ball die capable of adjusting the inside diameter during the reduction forming of aluminum tube body sections. The authors also identified the forming conditions required for successful die formation. Hirama et al. reported that tube polygonization is more likely to occur during forming when the diameter of the machining ball or the indentation amount is large [14]. This defect arises when the tube axis shifts from the machine axis owing to an increased forming load. Ou et al. [15] introduced a forming method for manufacturing triangular tubes, in which the machining tool rotates around the center of a circular tube. The results revealed that thickness reduction could be suppressed by optimizing the tool path, thereby improving formability. Zhu and Ji [16] were the first to propose a method for manufacturing spiral corrugated tubes using multiple rollers. Zhu and Ji [16] investigated the effects of various forming parameters on the forming load. The results indicated that forming temperature had the most significant impact and lubrication conditions had a minimal effect. Shi et al. [17] used numerical analysis to examine the effects of parameters such as roller tip radius and feed rate on formability in the hot forming of spiral corrugated tubes using multiple rollers. The authors [17] reported that feed rate and tensile stress contributed to damage during forming and recommended a forming temperature of 400 °C for 20G steel tubes. Although several studies have explored the manufacturing of spiral corrugated tubes using multiple rollers, most have relied on numerical analysis, with limited experimental investigations. Additionally, the length of parent tubes remains limited by the size of the forming machine. In conventional rotary methods, the parent tube is fixed in the machine and rotated, while the tool moves along the axis of tube. This results in limiting the maximum tube length to the machine’s size. By considering incremental forming methods [11,12] and rolling technology [18,19], a rotational forming method that feeds the parent tube instead of moving the tool is a promising solution. This approach maintains the benefits of rotary forming while removing the machine’s length restriction.

Therefore, this study presents a method for continuously manufacturing spiral corrugated tubes. In the proposed method, the rollers are fixed in the axial direction, and the parent tube is fed through the machine continuously. Compared to conventional forming methods, the proposed approach enables high-speed continuous production, significantly improves efficiency, and allows the length of the parent tubes to be freed from the limitations of the forming machine. Additionally, it minimizes downtime for tube removal. Several experiments and finite element analysis (FEA) were conducted to examine the effect of roller angle on formability and the forming mechanism. This study serves as a reference for designing continuous-production spiral corrugated tube forming machines and establishing optimal forming parameters. The proposed method directly reduces the energy consumption and carbon emissions. It also indirectly promotes the development of high-performance heat exchangers. The widespread use of heat exchangers equipped with spiral corrugated tubes is expected to further promote sustainable manufacturing across various industries.

2. Materials and Methods

2.1. Material

The samples were AA3003-H12 aluminum alloy tubes with an outside diameter of D₀ = 19.05 mm and an initial thickness of t₀ = 1.2 mm. AA3003-H12 is widely used in manufacturing heat exchangers owing to its excellent weldability and moderate strength. To evaluate the material properties of the parent tubes, tensile tests were conducted at room temperature (25 °C) using an Instron 5900R universal material-testing machine (Instron, Norwood, MA, USA). Tension specimens were cut from the parent tube according to JIS Standard No. 11 (Japanese Industrial Standards) [20]. A core was inserted to prevent the flattening of the clamping section during testing. The stress–strain diagram obtained from the tensile test is shown in Figure 2. The Young’s modulus (72 GPa) and Poisson’s ratio (0.29) were adopted from literature [21]. The material properties were extrapolated from the tensile-test results using an exponential work-hardening law for approximation.

2.2. Method and Conditions of Experiments and FEA

The experiments were conducted using the machine shown in Figure 3. The forming machine comprised three sections: the central forming section and the front and rear conveyors. Four rollers, which could revolve around the tube, were installed in the forming section. During the forming process, the tube was continuously passed through the forming section using a conveyer and the rollers were pressed against its outer surface to create spiral grooves.

In this study, complete understanding of the deformation behavior experimentally was challenging owing to the complex contact conditions between the rollers and the parent tube. FEA allows the visualization of stress changes and deformation processes under idealized conditions, which is beneficial for achieving a more detailed understanding of the forming process. Therefore, FEA was conducted to qualitatively examine the internal stress field and deformation behavior, rather than aiming high-precision quantitative reproduction. The effect of roller angle β on formability was investigated. The results were cross-validated with FEA, analyzing trends in terms of both forming defects and the dimensions of the formed tube. Table 1 and Figure 4 present the forming conditions and finite element model. This study investigated the effect of roller angle β using three angles: 32°, 36°, and 40°. The rotational speed n was increased from 130 rpm to 210 rpm at intervals of 20 rpm to determine the optimal angle β corresponding to the highest range of rotational speed n. To ensure reproducibility, the experiments were performed thrice under each condition. If a forming defect occurred in any of the three experiments, the condition was deemed unfavorable for forming and the defect was recorded. No lubricants were used in the experiments because the rollers were free to self-rotate.

The analytical conditions in the FEA were set to match the experimental setup. In this study, the model was created using Simufact Forming 2024.4, a commercial software developed by MSC Software Corporation (Newport Beach, CA, USA), and the implicit solution method was applied. For the FEA, instead of rotating the rollers, the parent tube was set to rotate around its own axis. The setup replicated the experimental conditions, where the free self-rotation of the rollers was caused by friction with the parent tube. Therefore, angular velocity was applied to the axis of the parent tube and the displacement degrees of freedom along the central axis were constrained at both ends of the tube. The mesh consisted of hexahedral elements, with 60 elements in the circumferential direction, 4 elements in the thickness direction, and 1 element per millimeter in the longitudinal direction. The Coulomb friction law was chosen to model the sliding behavior, where the friction force varies with the local contact pressure and area [22]. The friction coefficient μ was set to 0.05, as obtained from the literature [23]. This value is appropriate for the conditions of this study, which utilized freely self-rotating rollers without lubricants. The rollers were modeled as rigid bodies, whereas the tube was considered elastoplastic. This approach is justified because the rollers are made of quenched SKD11 steel (hardness ≈ 250 HB). Compared to AA3003 H12 (hardness ≈ 35 HB), the deformation of the rollers is extremely small within the forming load range of this study and can be considered negligible [24].

Table 1. Conditions for experiment and FEA.

Roller	Roller diameter DR [mm]	50
	Tip radius R [mm]	2.3
	Roller angle β [°]	32, 36, 40
	Indentation HI [mm]	2.4
	Rotational speed n [rpm]	130, 150, 170, 190, and 210
	Distance of moving LD [mm]	500, 123 (FEA)
	Speed of moving v [mm/s]	166.7
Tube	Initial diameter D0 [mm]	19.05
	Initial thickness t0 [mm]	1.2
	Initial length L0 [mm]	900, 150 (FEA)
	Material	AA3003 H12
	Element size (FEA)	1 mm/div along axial, 4 elements in thickness, and 60 elements in circumferential directions
Lubrication		None
Friction coefficient μ		0.05 [23]

Table 1. Conditions for experiment and FEA.

Roller	Roller diameter DR [mm]	50
	Tip radius R [mm]	2.3
	Roller angle β [°]	32, 36, 40
	Indentation HI [mm]	2.4
	Rotational speed n [rpm]	130, 150, 170, 190, and 210
	Distance of moving LD [mm]	500, 123 (FEA)
	Speed of moving v [mm/s]	166.7
Tube	Initial diameter D0 [mm]	19.05
	Initial thickness t0 [mm]	1.2
	Initial length L0 [mm]	900, 150 (FEA)
	Material	AA3003 H12
	Element size (FEA)	1 mm/div along axial, 4 elements in thickness, and 60 elements in circumferential directions
Lubrication		None
Friction coefficient μ		0.05 [23]

2.3. Evaluation of Formed Tube

As shown in Figure 5, thickness t, outside diameter D, peak pitch S, and groove depth h were measured to evaluate formability. Among these, t and D were the primary parameters, as they determine the feasibility of postprocessing and the pressure resistance of the product. t and D were measured from the cross-section of the formed tube. t was recorded at seven points (M1–M7) located at 15° intervals within the range of 0–90°. D was determined as the average of the 0° and 90° measurements. S and h were measured using a C3200 contracer (Mitutoyo Corporation, Kawasaki, Japan). The calculated value of S, S_cal, was determined by assuming that the tube length remained unchanged during formation. S_cal was calculated using the following equations:

$${\displaystyle \frac{s_{cal}}{\left(D-2H_I\right)\pi ∕4}}={\displaystyle \frac{v}{\left(D-2H_I\right)\pi \left(n∕60\right)}}$$

(1)

$$s_{\mathrm{c}\mathrm{a}\mathrm{l}}={\displaystyle \frac{60v}{4n}}$$

(2)

3. Deformation Mechanism and Tube Deformation at Different Roller Angles

This section examines the deformation process of the tube as inferred using FEA, providing an explanation of the tube-deformation mechanisms during forming. Subsequently, it discusses the effects of roller angle β and rotational speed n on the forming load P and deformation of the formed tube.

3.1. Deformation Mechanism in Cross-Section

Figure 6 shows the distribution of the outside radius D/2 (Figure 6a) and thickness t (Figure 6b) on the tangential cross-section of the tube during forming. These results were obtained under the conditions of β = 32° and n = 150 rpm. As the roller moved, the groove depth h increased, altering the t distribution. This change resulted from variations in the contact state between the roller and tube, as well as the material flow. As shown in Figure 7, when the roller is pressed onto the tube, the contact section of the material experiences compressive stress (Figure 7b). This stress causes the material in the contact section to flow into the noncontact sections on both sides of the roller (Figure 7c). Consequently, the thickness decreases at the groove bottom and increases on both sides, ultimately reaching the maximum thickness t_max on the rear side of the roller. (Figure 7d). Material flow primarily occurs at the outer surfaces of the tube, where direct contact with the rollers takes place. Interestingly, the material flows toward the rear side of the roller rather than the front side. This occurs because, in the observed tangential cross-section, the final contact is exclusively with the rear side of the roller.

Figure 8 illustrates the distribution of the outside radius D/2 (Figure 8a) and thickness t (Figure 8b) on the axial cross-section at roller angle β = 32° and rotational speed n = 150 rpm. As shown in Figure 8, the roller rotates counterclockwise relative to the tube. Similarly to the tangential cross-section, in the axial cross-section, thickness t on both sides of the roller increases as the roller is pressed into the tube. However, in the axial cross-section, the compressive stress on the front side is significantly greater than that on the rear side, causing material accumulation on the front side (Figure 9a). As forming continues, the rear side experiences tensile stress (Figure 9b). Finally, the maximum thickness t_max in the axial cross-section develops on the front side of the roller (Figure 9c).

In both the tangential and axial cross-sections, the material accumulates in the direction of the final contact section between the roller and tube. Owing to this accumulation, the thickness t increases to the maximum thickness t_max in each cross-section

3.2. Forming Load and Contact State

Figure 10 depicts the effects of roller angle β and rotational speed n on forming load P. The forming load P is divided into three directional components: radial load P_R, circumferential load P_C, and axial load P_A. Each component is positive in the centripetal direction, the direction of rotation, and the axial direction on the front side of the roller. As β increases, the load components P_R, P_C, and P_A tend to decrease. By contrast, P_R, P_C, and P_A tend to increase as n increases. This is consistent with previous multi-roller studies [16]. This trend is attributed to changes in the contact state between the roller and tube, prompting further investigation.

Table 2. Effect of roller angle β on contact state (FEA).

	β = 32°	β = 36°	β = 40°
130 rpm
150 rpm
170 rpm
190 rpm
210 rpm

presents the effects of β and n on the contact state. The contact length (C_L) and contact tip width (C_R) were analyzed to assess the differences in the contact states. As shown in Figure 11, for the same n, increasing β causes a decrease in C_L and an increase in C_R, leading to a reduction in P_R, P_C, and P_A. However, for the same β, C_L increases, and C_R decreases with the increase in n; consequently, P_R, P_C, and P_A. also increase. For the same value of n, energy efficiency can be improved by increasing β appropriately. Moreover, time efficiency can be further improved by increasing the moving speed v of the tube while keeping C_L and C_R constant.

3.3. Deformation of Tube at Different Roller Angles

The effects of roller angle β on the appearance of the formed tubes are shown in Figure 12. Additionally,

Table 3. Cross-section of formed tube.

	130 rpm	150 rpm	170 rpm	190 rpm	210 rpm
β = 32°
β = 36°
β = 40°

presents the cross-section at 40 mm from the edge of the formed section. As β increased, the range of rotational speed n corresponding to successful forming also expanded. When β = 40°, the tube was formed without defects, even at rotational speed n = 210 rpm. Therefore, β = 40° was determined to be the most suitable roller angle β for this study. The tube flattened during forming at β = 32°, n = 150 rpm and at β = 36°, n = 190 rpm. Figure 13 depicts the forming limits predicted by FEA, which closely align with the experimental values. In the FEA, the same forming defect occurs when β is small and when n is high. This is because large deformation leads to high instability in circumferential compressive deformation [14]. In this scenario, flattening occurs to eliminate excess circumferential lengths and loads.

4. Geometrical Precision and Circumferential Thickness Distribution

4.1. Geometrical Precision of Formed Tubes

The effects of roller angle β and rotational speed n on the peak pitch S are shown in Figure 14. S decreases as n increases, with the FEA and calculated values closely matching the experimental values (showing the same trend). However, for β = 32° and 36°, the experimental values were slightly higher than the FEA predictions. In contrast, for β = 40° the values were similar. The maximum discrepancy between the experimental and FEA values was 7.1%. As the number of waves remained constant for a given n, the observed error was likely because of tube length changes during the experimental forming process. For β = 32° and 36°, the formed tube tended to elongate, whereas at β = 40°, the tube length remained nearly unchanged.

Figure 15 illustrates the effect of roller angle β and rotational speed n on groove depth h. For β = 32° and 36°, the influence of n on h is minimal. A comparison of the experimental and analytical results reveals the same trend. However, for β = 40°, h decreases as n increases. The reasons for this trend are discussed later in this section. Additionally, the experimental h is smaller than the target value and the analytical h is larger. This discrepancy was likely because FEA was incapable of reproducing the spring-back phenomenon. During the forming process, the material in the formed groove sections experienced tensile stress (Figure 7). As n increased, elongation became more pronounced (Figure 14). Furthermore, as the tube ends were fixed during the forming process, the extra elongation caused h to exceed the target value. However, in the experiment, the tubes experienced spring-back after unloading, resulting in a smaller h than the target value. This effect became more prominent as n increased and the elongation intensified.

Figure 16 highlights the effects of roller angle β and rotational speed n on the outside diameter D. For each β, D generally decreases as n increases. However, at the same n, D tends to slightly increase with increasing β value. This difference was minimal: for n = 150 rpm, the variation between the values of D at β = 32° and 40° was only 0.1 mm. The experimental and FEA values followed the same trend, with a maximum error equal to 1.2%.

These results indicate that roller angle β and rotational speed n affect the contact state and peak pitch S, which together determine outside diameter D and groove depth h. For the same n, decreasing β leads to a decrease in D and an increase in S and h. This occurs because CL increases as β decreases (Table 2). The increased C_L causes the groove bottom to stretch more during deformation (Figure 7). Hence, despite the direct influence of n on S, the latter tends to increase as β decreases. Additionally, the radial forming load P_R and circumferential forming load P_C increase as C_L increases (Figure 11), reducing D and increasing h. By contrast, for the same β, both D and h decrease as n increases. As shown in Figure 17, S decreases with increasing n, causing the neighboring grooves to overlap during forming [16]. Consequently, D and h decrease, although C_L increases with increasing n. However, under small β and high n, C_L becomes excessively high and leads to forming failure (Figure 13).

Figure 18 depicts the effect of β on the equivalent plastic strain ε_e and equivalent stress σ_e distribution in the tangential cross-section at n = 150 rpm. As β increases, both ε_e and σ_e tend to decrease. For larger β values, deformation is reduced and local work-hardening is also suppressed. Therefore, at β = 40°, the quality of the formed product is expected to improve. However, the effect of β on the material flow in the tangential cross-section is considered minimal, as no significant local strain was observed in the ε_e distribution.

Figure 19 illustrates the effect of n on the equivalent plastic strain ε_e and equivalent stress σ_e distribution in the tangential cross-section. As n increases, ε_e also increases owing to a rise in the forming load P, caused by changes in the contact state (Figure 11). Therefore, n has a significant impact on material flow in the tangential cross-section, as higher plastic deformation occurs in localized areas. However, as n increases, σ_e initially rises but then decreases. This may be owing to the neighboring grooves compressing each other at higher rotational speeds (Figure 17), which neutralizes tension during forming (Figure 7).

Based on these results, n was identified as the primary factor influencing the tangential cross-sectional shape and material flow, while the influence of β was relatively minor. The prior studies also indicate that forming speed is the main factor [17,25]. Localized stress concentrations can be significantly reduced by selecting large β and low n. Additionally, FEA and experimental values showed good agreement for the outside diameter D and peak pitch S. Although the predictions from the proposed model showed slight deviations from the experimental values, the model remains valuable for early-stage research and industrial development, helping to reduce costs.

4.2. Circumferential Thickness Distribution

Figure 20 presents the experimental and FEA values for circumferential thickness t distributions under different conditions. Both values follow the same trend, showing that thickness t increases when the roller angle β is small and rotational speed n is high.

Figure 21 illustrates the average thickness t_ave and maximum thickness t_max obtained from both the experiment and FEA under different conditions. For the same β, both tave and t_max increase as n rises. This trend is attributed to local material accumulation, as material flow is enhanced with higher n. Conversely, at the same n, t_ave and t_max decrease as β increases. This is because a larger outside diameter D results in a smaller thickness t for higher β values. However, at β = 40°, the experimental t_max is smaller than the corresponding FEA value. This discrepancy is likely because of a measurement error, as accurately measuring t_max in experimental samples was challenging.

Figure 22 shows the experimental and FEA values for the thickness deviation ratio ϕ under various conditions. The thickness deviation ratio ϕ is obtained using the following equation:

$$\varphi ={\displaystyle \frac{t_{\mathrm{m}\mathrm{a}\mathrm{x}}-t_{\mathrm{m}\mathrm{i}\mathrm{n}}}{t_{\mathrm{a}\mathrm{v}\mathrm{e}}}},$$

(3)

where t_max, t_min, and t_ave are the maximum, minimum, and average thickness, respectively.

For β = 32° and 36°, the experimental and FEA values followed the same trend. As n increased, ϕ also increased, likely because of stronger material flow at higher n. Conversely, for β = 40° and n = 190 rpm or higher, the experimental ϕ values were lower than the FEA results. This discrepancy is attributed to measurement error in t_max. At β = 40° and n = 150 rpm, the minimum ϕ was observed, as t_max was also minimized under this condition. This suggests that t_max is influenced by material flow in not only the axial cross-section but also in the tangential cross-section. The conditions of β = 40° and n = 150 rpm resulted in the smallest axial load P_Z and the least material flow in the tangential cross-section. Except for β = 40° and n = 130 rpm, the thickness distribution becomes increasingly nonuniform as n increases. This is because the material rapidly undergoes substantial plastic deformation at high n. This can lead to increased strain localization, particularly on the front side of the roller (Figure 9). Conversely, as n decreases, the material deformation becomes gradual and uniform, resulting in improved thickness uniformity. This is consistent with reports that optimizing the roller path can improve thickness uniformity [26]. However, the effect of roller angle β was not significant across the range investigated in this study.

Figure 23 illustrates the effect of β on the distribution of the equivalent plastic strain ε_e and equivalent stress σ_e in the axial cross-section. As β increases, both ε_e and σ_e decrease. This is because a larger β results in a shorter contact length C_L and facilitates the self-rotation of the roller. However, no significant local strain is observed in the ε_e distribution, indicating that β has minimal effect on the material flow in the axial cross-section.

Figure 24 depicts the effect of rotational speed n on the distribution of equivalent plastic strain ε_e and equivalent stress σ_e in the axial cross-section. Both ε_e and σ_e increase with increasing n. This effect is particularly pronounced in the material flow on the outer surface.

5. Conclusions

This study investigated the effects of rotary forming parameters on the formability of spiral corrugated tubes via FEA and experiments. Compared to previous forming methods, the proposed method in this study successfully enables the manufacture of tubes without length limitation. The reproducibility and reliability of the results were confirmed by comparing the results of repeated experiments and the predictions of the FEA under the same conditions. The deformation behavior during the forming process was investigated. Finally, the relationship between the forming parameters and the dimensions of the formed tube was inferred from the deformation mechanism. In this study, a larger roller angle allows for a wider range of rotational speeds. When the roller angle was set to 40°, forming was achievable at a rotational speed of 210 rpm, making 40° the optimal roller angle in this study. This is because the contact states between the roller and tube varied with the roller angle. A small roller angle with a high rotational speed increased the contact length in the axial direction, leading to excessive circumferential load and tube flattening. In terms of forming tube dimensional accuracy, the rotational speed was the primary factor influencing the outside diameter and peak pitch of the tube, whereas the roller angle had minimal impact. By contrast, the groove depth was affected by both parameters, with a small roller angle and high rotational speed leading to increased groove depth. At a roller angle of 40° and a rotational speed of 150 rpm, the thickness deviation ratio was only 0.13, and the groove depth error from the target value was only 0.08 mm. These conditions enabled the precise formation of spiral corrugated tubes. In addition, material flow was greater on the outer surface than on the inner surface. A large roller angle and small rotational speed suppressed stress concentration and localized thickness accumulation, promoting uniform deformation.

However, this study is not without its limitations. The finite element analysis is qualitative. Consequently, the model includes several simplifications. In future work, we will develop high-accuracy FEA models and optimize the thickness uniformity and roller geometry.