Study on Incremental Sheet Forming Performance of AA2024 Aluminum Alloy Based on Adaptive Fuzzy PID Temperature Control

Abstract

The development of technology has driven a rising need for high-accuracy and high-efficiency manufacturing of low-volume products. Incremental forming technology, characterized by die-free flexibility and low production costs, can effectively replace stamping processes for manufacturing customized small-batch products. However, high-performance aluminum alloys generally exhibit poor room-temperature plasticity but excellent high-temperature plasticity, necessitating the integration of thermal-assisted methods for manufacturing such products. However, the temperature of the forming region will excessively rise without temperature control, which will affect the forming performance of the material in hot incremental sheet forming of AA2024-T4 aluminum alloy. This study focuses on AA2024-T4 aluminum alloy and proposes a uniform temperature control method for the electric hot tube-assisted incremental sheet forming process, incorporating an adaptive fuzzy PID algorithm. The temperature difference of the forming region is lower than 6% under the various temperatures. On this basis, the forming limit angle and the microstructure state of the material are analyzed, and the grain feature of the material exhibits significantly refined grains and the uniform fine grain distribution under 180 °C with the temperature control of the adaptive fuzzy PID algorithm.

1. Introduction

In recent years, due to the development of science and technology, the industrial product demand has been changed from small difference but large number of diversity with different requirements, etc., and the production mode has been changed from mass production to small-lot production [1,2]. To succeed, companies need to shorten time-to-market. Nevertheless, the conventional stamping fabrication process can be utilized in mass production, but the die development period is too long, and the production efficiency is low [3,4]. Thus, the traditional stamping technology cannot satisfy the need of contemporary high-precision batch customization [5,6]. Several flexible sheet-based forming processes had been developed to cope with this production paradigm [7,8]. The known mainstream forming technologies are incremental sheet forming (ISF), laser forming, ultrasonic forming and shot peening forming, etc. [9,10]. Among them, incremental sheet forming has drawn much attention and research due to its unique advantages [11,12]. The idea of flexible manufacturing was first proposed by Leszak [13] in the 1960s, and it has since been developed by Allwood et al. [14], developing into a general theoretical scheme. On this basis, Li et al. [15] systematically analyzed the influence of process parameters on the forming limit angle of AA2024 aluminum alloy using electrically heated incremental forming, identifying temperature as the primary factor enhancing this angle. Magnus et al. [16] and Mi et al. [17] applied direct current (DC) in forming tools, leading to a rapid temperature increase to 600 °C near tool-sheet interfaces, and the forming force had a substantial reduction. In addition to this, insulation devices are needed to avoid the current into machines and energy dissipation in Joule heating applications [18,19]. Meanwhile, Ti-6Al-4V titanium alloy showed the best forming performance at 500–690 °C, and the oxidation and the hydrogen embrittlement induced premature fracture and deteriorated formability at 700 °C [20,21]. Galdos et al. [22] improved the formability of AZ31B magnesium alloy through oil-bath heating of a sheet from the bottom to obtain an indirect heat transfer. Moreover, the oil-bath heating method is advantageous for adding backpressure in incremental forming. Local heating has the advantage of avoiding non-target deformation and increasing energy efficiency, but is very complicated to set the optimal parameters. To accurately match the forming paths with localized heat sources, Duflou et al. [23] proposed a laser-tool synchronous heating system, which heats the reverse side of the deformation zone, and it was used to form 1.25 mm thick AA5182 aluminum alloy thin-walled structures. Based on these, Lehtinen et al. [24] and Göttmann et al. [25,26] improved laser-assisted incremental forming systems, indicating that laser power influences significantly the formability and it must be adjusted with respect to material thermomechanical characteristics to obtain good parts. Meanwhile, Su et al. [27] found that the average grain size was stable in the step-down range of 0.2 to 0.7 mm, and the uniform grain size distribution was obtained at 0.7 mm step-down, and the significant coarsening of grain occurred at 1 mm step-down, with the increasing step-down decreasing formability.

Based on the above analysis, the temperature uniformity control, which will affect the forming performance of the material, remains a challenge in hot incremental forming. In this work, AA2024-T4 aluminum alloy is adopted to study the temperature uniformity regulation in hot incremental sheet forming. Based on the investigation results obtained previously [28,29], the electric hot tube heating method is used to generate the target components in hot ISF, as presented in Figure 1. The distribution law of temperature gradients is studied in the forming region. The temperature control system offers the possibility of online monitoring of the forming region temperatures in order to keep the sheet in the ideal forming temperature range so as to improve the formability of AA2024-T4 aluminum alloy.

2. Materials and Methods

An AA2024-T4 alloy sheet, with a size of 200 mm × 200 mm and a thickness of 1.0 mm, was adopted to fabricate the part. The chemical composition of AA2024-T4 alloy is shown in

Table 1. Chemical composition of AA2024-T4.

Al	Si	Fe	Cu	Mn	Mg	Cr	Zn	Ti
Balanced	0.5	0.5	3.8	0.3	1.2	0.1	0.25	0.15

. A high-temperature chain oil of 600 °C is used to ensure the surface quality of materials in this work.

To analyze the forming limit angle, a variable-angle conical part of AA2024-T4 aluminum is adopted in this work. A radius fillet of 5 mm is designed at the opening position of parts to ensure the smooth transition of the forming path. The 3D model of the forming part is shown in Figure 2a, and the model, as shown in Figure 2b, includes an opening radius of 60 mm, a bottom radius of 15 mm, a forming depth of 50 mm, and a curvature radius of 70.94 mm. The forming angle is in the range from 19.71° to 76.32°.

To determine the maximum forming limit angle of AA2024-T4 aluminum alloy after incremental sheet forming, negative incremental sheet forming tests are conducted using the variable angle cone. The corresponding experiments, as shown in

Table 2. The experimental scheme.

No.	Tool Diameter (mm)	Step Depth (mm)	Feed Rate (min/mm)	Temperature (°C)	Temperature Control Status
1	16	0.2	1500	25	off
2	16	0.2	1500	120	no
3	16	0.2	1500	150	on
4	16	0.2	1500	180	off
5	16	0.2	1500	180	no
6	16	0.2	1500	210	no

, are conducted under varied heating temperatures (25 °C, 120 °C, 150 °C, 180 °C, 210 °C) and temperature control modes when the other process parameters are constants, such as 16 mm tool diameter, 0.2 mm step depth, and 1500 mm/min feed rate.

A forming device coupled with temperature acquisition, as shown in Figure 3, is adopted to obtain forming temperature and manufacture parts according to the previous research result [15]. Four hot pipes (Single power: 500 W) are adopted to heat the edge of the forming region through the contact heat transfer method, and the heat flows from the edge to the center of the forming region, which causes the temperature rise in the forming region. The hot-working die steel of H13, which had good strength at high temperatures, is used to fabricate the clamp and support plates. The mica is adopted to fabricate insulating plates. Meanwhile, the use of insulating plates could limit the heat loss at the edge. The power regulator is used to control the power of each heating tube. Meanwhile, K-type temperature sensors (range: 0–1300 °C) are used to obtain temperature signals of the forming region, and the multichannel temperature collector is adopted to obtain the temperature value of the forming region. The use of K-type temperature sensors can calibrate the emissivity to improve the acquisition accuracy of the thermal imager. In addition to this, a thermal imager (Manufacturer: FOTRIC, Santa Clara, CA, USA; Model: 700HD; Range: −20 °C to 1300 °C; Error: ±1 °C) is adopted to collect temperature for the forming region, and the emissivity of 0.02 is set up through the calibration of K-type temperature sensors. Meanwhile, the testing instrument (Manufacturer: AMETEK, Berwyn, PA, USA; Model: EDAX Hikari Plus) of the electron backscatter diffraction (EBSD) is adopted to analyze micro-characteristics of the material, and the grain size is analyzed through EDAX Hikari Plus. The EBSD data are collected based on a slice step size of 0.07 μm, a 2 × 2 pixel binning mode, and 30 kV accelerating voltage with an electron beam current of 5 nA.

The electric hot tube assisted temperature control system is adopted to control the forming temperature in the hot incremental sheet forming process of AA2024-T4 aluminum alloy to ensure that. The forming temperature fluctuates within a small range of the design value. To regulate the temperature control system, it is necessary to continuously monitor the temperature of the forming region and to process the thermal data within the controller using control algorithms that keep the forming temperature in the designed range. Therefore, the temperature control system needs to have some features, such as running steadily, less overshoot, quick response, and less static error. The electric hot tube assisted temperature control system primarily consists of electric heating tubes, temperature sensors, a central controller, a heating circuit, and solid-state relays. Real-time temperature data from the sheet-forming region is collected by temperature sensors and transmitted to the central controller. Processed and embedded by a temperature algorithm, the controller, as shown in Figure 4, outputs a pulse-width modulation (PWM) signal to control the switching frequency of the high-frequency power supply and to change the average heating power and to realize accurate temperature control in the forming region.

According to the previous study [28], the temperature between the clamp region and the forming region has a gradient, and the temperature model is written as:

$$T\left(x\right)={\displaystyle \frac{T_2^{\prime }\left(t\right)+\lambda u_1-\lambda T_2\left(t\right)}{\lambda x+\alpha }}x+T_2\left(t\right)$$

(1)

where T(x) is the temperature function about x. x and t are separately the distance between the clamp region and the forming region, and the heating time. Meanwhile, α is equal to k₁/c₁ρ, and λ is equal to 2k₂/c₁ρ, in which k₁, c₁, k₂, and ρ are separately the thermal conductivity of the sheet, the specific heat of the sheet, the surface thermal conductivity between the sheet and air, and the density of the sheet. u₁ is the temperature of the air, and it is equal to the room temperature. In addition to this, the clamp region temperature (T₂(t)) and the temperature rise rate of the clamp region (T₂(t)) are written as:

$$\left\{\begin{array}{c}{T}_2(t)=3.37P^{0.98}{e}^{-7.118\times {10}^{-7}t}-3.25P^{0.99}{e}^{-0.00079t}\\ T_2^{\prime }\left(t\right)=2.57\times {10}^{-3}{P}^{0.99}{e}^{-0.00079t}-23.99\times {10}^{-7}{P}^{0.98}{e}^{-7.118\times {10}^{-7}t}\end{array}\right.$$

(2)

where P is the power of heating pipes. In this work, the heat parameters of the sheet are separately 0.234 W/mm °C of conductivity, 900 J/kg·°C of specific heat, and 2.7 × 10⁻⁶ kg/mm³ of density. The surface thermal conductivity between the sheet and air is 2.5 × 10⁻⁵ W/mm²·°C, and the room temperature is 10 °C, and the range of the power is 0 to 2000 W. According to the structure of the forming part, the temperature difference between the clamp region and the forming region is approximately 1–2 °C within the range of 200 °C. Therefore, the effect of the gradient on the forming temperature can be ignored, and the clamp region temperature can be viewed as a control target to realize the temperature control in the forming region.

The algorithm research of temperature control systems mainly includes traditional PID, fuzzy control, and adaptive fuzzy PID control algorithms. In traditional PID control, the proportional, integral, and derivative parameters are tuned once and cannot be modified online. In actual control processes, when the system is disturbed, the performance of fixed PID parameters is poor and may result in significant deviations. Fuzzy control systems may involve large amounts of computation during fuzzification, fuzzy inference, and defuzzification, resulting in high computational load and poor real-time performance, especially in multiple-input and multiple-output (MIMO) systems. In order to resolve this issue, the fuzzy control algorithm is combined with the traditional PID algorithm to form an adaptive fuzzy PID control method [30]. PID control is suitable for systems with accurately known mathematical models. The system input is viewed as the deviation between the actual output and the desired output, and the combination of proportional, integral, and derivative is adopted to calculate the control quantity and to regulate the controlled system, and the principle of PID control is shown in Figure 5.

According to Figure 5, the dashed box indicates the PID linear controller. r(t) denotes the desired output, and y(t) is the actual output. Meanwhile, the system error e(t) is defined by the relationship:

$$e\left(t\right)=r\left(t\right)-y\left(t\right)$$

(3)

where t is the time. The system error e(t) is processed by the combined calculation of the PID linear controller. Meanwhile, the control signal u(t) applied to the controlled object can be obtained, and its expression is given by the following equation:

$$u(t)=K_P\left[e(t)+{\displaystyle \frac{1}{T_i}}{\int }_0^{t}e(t)dt+{\displaystyle \frac{T_{d}de\left(t\right)}{dt}}\right]$$

(4)

Based on Equations (3) and (4), the resulting transfer function can be obtained, and it is written as:

$$G(t)={\displaystyle \frac{u\left(t\right)}{e\left(t\right)}}=K_P\left(1+{\displaystyle \frac{1}{T_{i}t}}+T_{d}t\right)$$

(5)

where K_P is the proportional gain, and T_i is the integral time constant, and T_d is the derivative time constant.

In the adaptive control algorithm of fuzzy PID, the three items of proportional, integration, and differentiation can be adapted to the actual working conditions for self-optimization, thus achieving better control effects, and the corresponding principle is shown in Figure 6. System input parameters are the error and the error change of actual and reference temperature values. After fuzzification of the input parameters, the fuzzy controller performs fuzzy inference and defuzzification to determine the adjustment values of the PID parameters, such as ΔK_P, ΔK_i, and ΔK_d. By adding the adjustment values to the parameters of the traditional PID algorithm, the parameter adaptation process is achieved. Subsequently, the updated PID parameters are used to control the system.

The output of the adaptive fuzzy PID controller includes ΔK_P, ΔK_i, and ΔK_d with real physical domains of [−10, 10], [−0.8, 0.8], and [−5, 5], respectively, while the fuzzy domain range is [−6, 6]. Taking ΔK_P as an example for defuzzification, the processing method can be expressed as:

$$\left\{\begin{array}{c}{K_P}^{*}={\displaystyle \frac{{\sum }_{i=1}^{49}\mu \left(K_{P_i}\right)·C_{P_i}}{{\sum }_{i=1}^{49}\mu \left(K_{P_i}\right)}} \\ {∆K}_P={\displaystyle \frac{{K_P}^{*}·(K_{P_H}-K_{P_H})}{x_H+x_L}}+{\displaystyle \frac{K_{P_H}-K_{P_H}}{2}}\end{array}\right.$$

(6)

where K_P^* is the precise value in the fuzzy domain, and μ(K_Pi) is the membership degree of the proportional coefficient in the output variable, and C_Pi is the value of the corresponding element in the fuzzy output.

The calculation formula for the system control parameters can be written as:

$$\left\{\begin{array}{c}{K}_P=K_p^{\prime }+∆K_p\\ K_i=K_i^{\prime }+∆K_i\\ K_d=K_d^{\prime }+∆K_d\end{array}\right.$$

(7)

where K_P, K_i, and K_d are the final PID control parameters. K_p′, K_i′, and K_d′ are the PID parameters of the traditional PID control algorithm.

A simulation model of the adaptive fuzzy PID algorithm is established in Simulink, as shown in Figure 7. The inputs are e and e_c with quantization factors of 0.6 and 1.2, respectively. The outputs are ΔK_P, ΔK_i, and ΔK_d with quantization factors of 0.6, 7.5, and 1.2, respectively. The PID initial values are set as the typical PID values, and the simulation results of the adaptive fuzzy PID algorithm are plotted in Figure 8. The simulation results indicate that the adaptive fuzzy PID algorithm has tiny overshoot and steady-state errors, and the settling time is 100 s. Therefore, the use of the adaptive fuzzy PID controller can improve the dynamic performance of the temperature control system.

3. Results and Discussion

3.1. Analysis of Control Algorithms

Figure 9 shows the simulation results of traditional PID, fuzzy control, and adaptive fuzzy PID control. A comparison of the three simulation results shows that the use of the traditional PID control tends to cause a large overshoot in the controlled system. Although the single use of fuzzy control can reduce the overshoot of the system, it results in a long settling time and slow response speed. On this basis, the adaptive fuzzy PID control is obtained by combining fuzzy control with traditional PID control, which reduces the system overshoot and shortens the settling time. Therefore, the adaptive fuzzy PID control is adopted as the control algorithm for the temperature control system in hot ISF with electric hot tube heating.

The comparison of actual and simulated temperatures under the control effect of adaptive fuzzy PID control is shown in Figure 10. The central temperature of the forming region is 150 °C, and the temperatures of the other four positions are separately 152 °C, 157 °C, 164 °C, and 172 °C, and the temperature difference of the forming region is 7 °C. Meanwhile, the temperature gradient of the forming region is approximately 5 °C with the simulation. When the center temperature is 180 °C, the temperature difference in the forming region is 10 °C that is similar to the simulated result. In addition to this, the temperatures of the other four regions are 214 °C, 221 °C, 230 °C, and 241 °C under the center temperature of 210 °C, and the temperature gradient of the forming region is 11 °C, and the corresponding simulated difference is 10 °C. In summary, the differences between the experimental and simulated temperatures are both less than 2 °C, and the temperature difference of the forming region is lower than 6% under the various temperatures. According to the above analysis, the temperature of the forming region meets the processing requirements of AA2024-T4 aluminum alloy, and then the adaptive fuzzy PID control method can realize the uniformity of the forming region temperature in hot ISF.

3.2. Analysis of Forming Limit Angles

During the incremental forming process, the tool presses the sheet along the forming path. When a sharp noise, namely that which is a mark of the initial fracture, is obtained, the forming process is immediately halted, and the forming depth is measured through the height ruler. Based on the curvature radius (R) and the forming depth (h_C), the forming limit angle is calculated, and the corresponding formula is written as:

$$\alpha ={\mathit{cos}}^{-1}\left({\displaystyle \frac{h-h_C}{R}}\right)$$

(8)

where α is the forming limit angle, and h is the designed depth of the part.

According to the aforementioned experimental scheme, the forming limit angle of each experimental group is calculated based on Equation (8), and the corresponding result is shown in Figure 11. A comparative analysis between No.4 and No.5 was conducted to evaluate the effect of the temperature control on the forming limit angle. Meanwhile, the effect of the different temperatures on the forming limit angle is further analyzed under the action of the adaptive fuzzy PID control. The comparative tests are conducted with a heating power of 1200 W, and the group without the temperature control obtains a forming temperature of 310 °C and the designed temperature is 180 °C. The excessively high forming region temperature is obtained due to the fact that the electric hot tube continuously supplies heat to the sheet. In this case, the material would obtain an excessive softening phenomenon, and the friction resistance between the forming tool and the sheet surface would increase, which would lead to the premature fracture of the part. In addition to this, the change of the forming limit angle is analyzed with different forming temperatures and the adaptive fuzzy PID control. From 25 °C to 180 °C, the forming limit angle gradually increases with rising temperature. When the temperature reaches 210 °C, the forming limit angle decreases to some extent due to excessive softening of the material, which intensifies the surface wear (shown in Figure 12) and the thickness reduction, and the premature fracture of the part is obtained.

3.3. Analysis of Microstructure

According to the previous study, the grains of AA2024-T4 aluminum alloy gradually grow with the increase of 120 °C to 210 °C, and the dynamic recovery and the dynamic recrystallization separately occur at 180 °C and 210 °C. On this basis, to investigate the micro-mechanism under the action of the temperature control, square specimens of the same size were sampled from the sidewall region of the parts, which were obtained through hot ISF, to analyze the material microstructure. The material microstructures, as shown in Figure 13 with a scale bar of 100 μm, are obtained under various forming conditions, such as room temperature (Figure 13a), 180 °C with temperature control (Figure 13b), and 180 °C without temperature control (Figure 13c). Compared to the microstructure at room temperature, the grain feature of the material exhibits significantly refined grains and the uniform fine grain distribution under 180 °C with temperature control. The increase in the uniform temperature results in microstructure changes, such as even heating of atoms within the grains, improvement of stacking fault energy, and uniform dynamic recrystallization. As deformation increases, the material flows more easily due to the uniform temperature, and then the forming limit angle of the material is improved. However, the grains of the material are provided with alternating regions of refinement and coarsening under the action of 180 °C without temperature control. The edge heat of the forming region will flow towards the center of the forming region under the action of a temperature control system, which results in a small temperature gradient (Figure 14a) that can be ignored in the forming region. However, the edge temperature of the forming region will continue to rise without a temperature control system, and the increase rate is higher than the heat conduction rate, which is resulting in a significant temperature gradient (Figure 14b) in the forming region. The forming temperature is gradually increased with the increase in forming time due to the fact that the temperature control is not adopted, and then the excessive forming temperature and the significant temperature gradient are obtained. Therefore, the dynamic recrystallization is obtained in the low-temperature region, and the grain growth, which leads to the uneven thermal stress distribution and the reduction of the forming limit angle, is acquired in the high-temperature region.

4. Conclusions

Based on simulation and experimental analysis, three temperature control algorithms, such as conventional PID, fuzzy control, and adaptive fuzzy PID, were used to analyze the overshoot and steady-state errors of the temperature control in the forming region during incremental sheet forming with electric hot tube heating. Based on the adaptive fuzzy PID algorithm, the tiny overshoot, namely a short settling time of approximately 100 s, and the steady-state error are obtained in the temperature control system, which is beneficial for the precise temperature control. The control accuracy of the algorithm was validated through temperature measurement experiments during hot incremental sheet forming. Meanwhile, the effect of different heating temperatures and temperature control conditions on the forming limit angle of the parts was also investigated. The specific conclusions are as follows:

• The material forming region obtains a uniform temperature distribution and better forming performance with temperature control. In contrast, the excessive heating leads to severe material softening and increased wear without temperature control, and then the forming performance is weakened.
• The forming limit angle gradually increases with the increase in the forming temperature in the range of 25–180 °C. When the temperature reaches 210 °C, the forming limit angle decreases to some extent due to excessive softening of the material, which leads to the surface wear and the thickness reduction, and the premature fracture of the part is obtained.
• The effect of the temperature control on the microstructure of the material was studied. The forming region obtains a uniform fine grain distribution at 180 °C with temperature control, and the alternating fine and coarse grains are obtained without temperature control, which leads to the uneven stress distribution and the decrease in the forming performance.

In summary, the material achieves a positive plastic deformation ability and favorable microstructural characteristics at 180 °C with the adaptive fuzzy PID control. In future research, the quantitative relationship between the geometric accuracy and the adaptive fuzzy PID control needs to be further investigated in detail. Meanwhile, the corresponding result can be further extended and applied to the forming of other grades of aluminum or magnesium alloys during hot incremental sheet forming. In the future, the association model between forming process parameters and the temperature control algorithm needs to be further established, and the online optimal regulation of the forming temperature should be studied in detail in hot incremental sheet forming.