Modelling of the effects of process parameters on energy consumption for incremental sheet forming process

Journal Pre-proof Modelling of the Effects of Process Parameters on Energy Consumption for Incremental Sheet Forming Process

Fuyuan Liu, Xiaoqiang Li, Yanle Li, Zijian Wang, Weidong Zhai, Fangyi Li, Jianfeng Li PII:

S0959-6526(19)34326-4

DOI:

https://doi.org/10.1016/j.jclepro.2019.119456

Reference:

JCLP 119456

To appear in:

Journal of Cleaner Production

Received Date:

05 June 2019

Accepted Date:

24 November 2019

Please cite this article as: Fuyuan Liu, Xiaoqiang Li, Yanle Li, Zijian Wang, Weidong Zhai, Fangyi Li, Jianfeng Li, Modelling of the Effects of Process Parameters on Energy Consumption for Incremental Sheet Forming Process, Journal of Cleaner Production (2019), https://doi.org/10.1016/j. jclepro.2019.119456

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Modelling of the Effects of Process Parameters on Energy Consumption for Incremental Sheet Forming Process Fuyuan Liu a, b, 1, Xiaoqiang Lic, d, e, 1, Yanle Lia, b *, Zijian Wang a, b, Weidong Zhai a, b, Fangyi Lia, b, Jianfeng Li a, b a

Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education,

School of Mechanical Engineering, Shandong University, Jinan 250061, China b

National Demonstration Center for Experimental Mechanical Engineering Education (Shandong

University) Jinan 250061, China c

School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China

d

Beijing Key Laboratory of Aircraft Assembly & Robotics Technology, Beihang University,

Bejing 100191, China e

Innovation Centre of Advanced Manufacturing, Ningbo Institute of Technology, Beihang

University, Ningbo 315100, China *Corresponding

1:

author: [email protected]; Tel: +86-0531-88392208

First Authors

Abstract Incremental sheet forming (ISF), as a flexible rapid prototyping technology, has great potential in the production of small-volume complex sheet parts. The research on the energy consumption of ISF is beneficial to the determination of the most energy-saving process parameters. First, the total power of the machine tool is broken down into standby power, feed axis power and sheet forming power, which is also theoretically analyzed, respectively. Apart from the modeling of the standby power and feed axis power, a theoretical mechanism model for sheet forming power during the ISF process is established based on the contact area and the flow condition of sheet. Then, experiments at the standby state, idle feed state, air forming state and actual processing state are carried out respectively to determine the essential coefficients of the theoretical model. In addition, the processing power prediction model in ISF is obtained and the prediction accuracy is verified through experiments. The results confirmed that the power prediction error of the processing power is below 5%. Moreover, the effects of process parameters (forming tool radius, step down, sheet thickness, feed rate) on processing power, power efficiency, processing energy and energy efficiency are comprehensively analyzed. Finally, the optimal combination of process parameters for the lowest energy consumption is obtained. Keywords: Incremental sheet forming; energy consumption; power model; contact area Nomenclatures dr

The projected length of the contact line in the feed direction in the XOZ plane

Journal Pre-proof 𝐸𝐹𝑒𝑒𝑑 𝐸𝑃𝑟𝑜𝑠𝑒𝑠𝑠 𝐸𝑆ℎ𝑒𝑒𝑡 𝐸𝑆𝑡𝑎𝑛𝑑 𝐸𝑈𝑠𝑒 𝑙 𝐿 𝐿' 𝑃𝐴𝑖𝑟 𝑃𝐶𝑜𝑛𝑡𝑟𝑜𝑙

𝑃𝑑 𝑃𝑓 𝑃𝐹𝑎𝑛 𝑃𝐹𝑒𝑒𝑑 𝑃𝐿𝑢𝑏

𝑃𝑀acℎ𝑖𝑛𝑒 𝑡𝑜𝑜𝑙 𝑃𝑆ℎ𝑒𝑒𝑡 𝑃𝑃𝑟𝑜𝑐𝑒𝑠𝑠 𝑃𝑆𝑝𝑐𝑜𝑜𝑙 𝑃𝑆tand 𝑃𝑈𝑠𝑒 𝑟𝑡𝑜𝑜𝑙 t 𝑡0 𝑣𝑓 𝛼 𝛽 𝜔 𝛾 𝜑 𝜎t

Energy used for feed axis only Energy consumed during the whole process Energy used for sheet deformation only Energy consumed by standby unit or during standby state Energy used to deform the sheet, sum of 𝐸𝑆ℎ𝑒𝑒𝑡 and 𝐸𝐹𝑒𝑒𝑑 Length of motion track at a point on the sheet between two adjacent passes The friction curve length of the point at the 𝛾 position The projected length of the two end points of 𝐿 in the Y axis Total machine power measured under air forming state Power consumed by control system Deformation power of sheet, one part of 𝑃𝑆ℎ𝑒𝑒𝑡 Friction power between forming tool and sheet, another part of 𝑃𝑆ℎ𝑒𝑒𝑡 Power for electric fan Power for feed axis working only Power for basic lubrication cooling device Total machine tool power. Power for sheet deformation only Total machine power measured under processing condition

Power for spindle forced lubrication cooling device Total machine power measured under standby condition(equals to standby power) Power used to deform the sheet, sum of 𝑃𝑆ℎ𝑒𝑒𝑡 and 𝑃𝐹𝑒𝑒𝑑 The radius of the forming tool head Actual sheet thickness in a certain forming position Initial sheet thickness Feed rate Forming angle Contact angle (the angle of the contact arc between the tool head and the sheet in the YOZ plane) The corresponding central angle of the contact line in the feed direction The central angle corresponding to the arc between the tangent points and a certain point in XOZ plane Parameter of cycloid equation Through-thickness stress under the forming tool

Journal Pre-proof 𝜎𝑌 ∆𝑧 𝜏𝜃 𝜓 𝜙 𝜇𝜃

Yield strength Step down size Circumferential shear stress under the forming tool Power efficiency Energy efficiency Coefficient of friction

1 Introduction Reducing the energy consumption during the manufacturing process has become an important concern of sustainable development. To deform the sheet, machine tool consumes electricity. According to International Energy Agency (IEA), more than half of the electricity were generated from non-renewable resources such as coal, oil and natural gas in 2018. Hence, the consumption of electricity indirectly produces carbon dioxide, which aggravates the greenhouse and has a direct impact on the environment. In the European Union, the energy consumption of the machine tool not only characterizes the environmental performance of the product but also reflects the environmental impact of the manufacturing process (Zhou et al., 2017). The modeling of energy consumption contributes to the determination of the optimal process parameters to save processing energy, while ensuring processing quality and reducing processing time. Incremental sheet forming (ISF) is a flexible forming technology. In ISF process, the sheet is squeezed layer by layer with a rigid forming tool with a hemispherical head. Then the local plastic deformation is accumulated to form the desired shape. As shown in Fig. 1(a), the sheet is fixed on the machine table by the fixture. The forming tool, as shown in Fig. 1(b), moves along the designed trajectory to deform the sheet to the designed shape (Fig. 1(c)). Since molds are not essential in ISF, the production cycle is shorter and cost is lower comparing with conventional stamping. In addition, ISF has the advantages of enhanced forming formability, smaller forming force and flexible shape geometry. However, the ISF is still not widely used in industrial production due to unsatisfactory forming precision and imperfect surface quality. Scholars are committed to conduct fundamental research on ISF and promote the commercialization of ISF. Energy consumption research plays a vital role in saving energy and reducing carbon emissions, which has become one of the important research topic for ISF.

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Fig. 1. The principle of incremental sheet forming process (a) ISF principle (Ingarao et al., 2012), (b) Forming tools, (c) Deformed workpiece

Most scholars focused on the energy consumed by the machine tool during the ISF process. Ambrogio et al (Ambrogio et al., 2014) carried out ISF experiments on a CNC milling machine and a CNC lathe and claimed that high-speed ISF experiments conducted on the lathe effectively reduced the forming time and improved the energy efficiency. Meanwhile, Ingarao et al. (Ingarao et al., 2014) studied the life cycle list, the processing time allocation, the total electric power, the total energy consumed and the electric energy consumed by each unit when ISF experiments were conducted on a CNC milling machine and a six-axis robot respectively. To decrease the energy consumption, Duflou et al. (Duflou et al., 2012)proposed three strategies to improve the energy and resource efficiency including: (a) redesign of machine tools and selective modes control, (b) distribution of machine tools by power grade, and (c) optimization of process parameters. Some scholars studied the sheet forming energy and heating energy. Both Li et al. (Yanle Li et al., 2015)and Yao et al. (Yao et al., 2017) used the response surface method to study effects of process parameters on sheet forming energy consumption and forming quality. Li et al. (Yanle Li et al., 2015) obtained the best combination of parameters for improving both energy efficiency and geometric accuracy. Yao et al. (Yao et al., 2017) obtained the best optimal parameter combination for energy saving, surface roughness and geometric accuracy. Moreover, the study of Ambrogio et al. (Ambrogio et al., 2012) indicated that the heating efficiency of joule heating system (around 53%) was much higher than that of laser heating system (around 10%) when a titanium sheet was processed during hot incremental sheet forming. Meanwhile, joule heating system had simpler structure. Some scholars not only studied the energy consumption of ISF, but also studied its environmental impact. Bagudanch et al. (Bagudanch et al., 2016) investigated the

Journal Pre-proof environmental costs of ISF with polymer materials, and a processing guide based on environment effect for ISF was developed. Dittrich et al. (Dittrich et al., 2012) simulated the environmental impact of single point incremental forming, double-point incremental forming, hydroforming and conventional forming methods through exergy analysis method. The results showed that the decrease of lubricant and electricity consumption during unload condition led to higher efficiency. Branker et al. (Branker et al., 2012) firstly analyzed the cost, energy and carbon dioxide emissions in a SPIF process for manufacturing a custom designed aluminum hat. By doubling the feed rate and step-down size, as well as using an eco-benign lubricant, it was found that the cost and energy consumed during the process without labor reduced from $4.48 to $4.10 and 4580 kJ to 1420 kJ, separately. Then, they (Branker et al., 2013) further investigated sustainability issues in SPIF by estimating the associated CO2 emissions and calculating the related costs. In addition, some scholars also compared the energy consumption of different processes. Ingarao et al. (Ingarao et al., 2011) established a model to compare the traditional stamping and ISF from the perspective of sustainability, in which the waste of sheet metal was considered. Later, Ingarao et al. (Ingarao et al., 2012) proposed an analysis method between environmental cost and environmental burden to evaluate the environment impact of different processes. Cooper et al. (Cooper et al., 2017) also compared the energy consumption of different processing technologies. In particular, power consumptions of five sheet forming processes (hydraulic drawing press, stretch forming, fluid cell forming, superplastic forming and ISF) were studied and their environmental impacts were analyzed. Moreover, Cooper et al. (Cooper and Gutowski, 2018) proposed a methodology to assess the positive impact of the double-sided incremental sheet forming. The results showed that around 100 TJprimary and 60 million U.S. dollars per year by 2030 would be saved when use double-sided ISF in the U.S. car industry. Petek et al. (Petek et al., 2007) compared SPIF and deep drawing as well on economic costs. The results indicated that SPIF process was more suitable for small batch production while conventional deep drawing was more economical when the production amount was larger than 625. In summary, most researches were only the analysis and inference of the result of a set of experiments or the simulation, such as the research of Ambrogio et al. (Ambrogio et al., 2014) on the machine tool energy consumption, the research of Ambrogio et al. (Ambrogio et al., 2012) on the heating energy, the research of Ingarao et al (Ingarao et al., 2011) on the environmental, etc. Several researchers established the empirical models with certain universality, such as the research of Ambrogio et al. (Ambrogio et al., 2014) on the energy consumption of a six-axis robot, the study of Li et al. (Yanle Li et al., 2015) and Yao et al. (Yao et al., 2017) on effect of process parameters on sheet forming energy consumption and forming quality. However, empirical models did not correlate well with phenomena and models, and the use was limited. Therefore, it was necessary to propose a universal theoretical model to solve the energy consumption prediction of ISF.

Journal Pre-proof In terms of energy modeling and optimization of the machine tool, the research was relatively mature. Li et al. (Li et al., 2011) pointed out that the fixed energy consumption of machine tool (the standby power) occupied a considerable proportion of the total energy consumption after analyzing six different CNC machine tools. Gutowski et al. (Gutowski et al., 2005)and Altintas et al. (Altintas et al., 2011) also claimed that the fixed energy consumption of the machine tool was much larger than the energy consumed by material processing. Once the machine tool was idle, it consumed a large amount of invalid energy. In the aspect of feed axis power, (Altintas et al., 2011) et al. reviewed the design and control of the machine feed drive system and analyzed the machine tool guides designed by friction, rolling original, static pressure and magnetic suspension principle. Moreover, Edem et al. (Edem and Mativenga, 2016) proposed a new CNC machine feed axis power model which takes into account the weight of the feed axis and the influence of the material. Kim et al. (Kim et al., 2015) presented a simple methodology for capturing the power consumption by accelerating and decelerating of the feed drive on a vertical three-axis milling machine and concluded that the peak power and additional energy consumption by feed rate can be estimated by a 2 order polynomial according to change of feed rate. Recently, Zhou et al. (Zhou et al., 2018) proposed an optimization model focusing on minimizing the process time and unit energy consumption (UEC). It was indicated that proper parameter combination could effectively minimize the processing time and UEC. In short, scholars have performed substantial researches on energy consumption of the commonly used machine tools. Since the ISF process can be also carried out on the similar machine tool, it is reasonable to apply the same energy analysis method to establish the energy consumption model of ISF process. Nevertheless, the relation between theoretical ISF energy model and machine tool electric energy consumption model has not received considerable attention. Based on the experimental observation, Asghar et al. (Asghar et al., 2014) approximated the contact area between the forming tool and the sheet as a rectangle in ISF, and the instantaneous force between the plate and the forming tool during process was calculated based on the contact area, as shown in Fig. 2(a). Bansal et al. (Bansal et al., 2017) further confirmed the shape of the contact area and then improved the contact area model. They assumed the contact area as a part of sphere, as shown in Fig. 2(b). The forming force prediction model of ISF and MPIF were established. Finally, the verification experiments of different process parameters were carried out which showed that the prediction accuracy of the model was obviously increased. Recently, Chang et al. (Chang et al., 2019) divided the contact section into three sub-areas to calculate the forming force of three typical incremental forming processes (SPIF, MPIF and IHF). Based on the contact area, Silva et al.'s membrane analytical method (Silva et al., 2008) was improved to calculate the forming forces of the three processes. The results showed that the prediction model had higher accuracy. In ISF process, although the contact area between the sheet and the forming tool

Journal Pre-proof plays an important role to predict the forming force and sheet forming power, the theory to calculate the contact area was still not adequate. First, the current geometry of the contact area was established based on the supposed contact condition, rather than the actual observed contact condition. Second effects of parameters on the contact area were not researched in detail. As a result, a complete set of theories should be proposed to describe, analyze and apply the contact area between forming tool and sheet in ISF process.

Fig. 2. Contact area during incremental sheet forming process (a) Contact area calculation (Asghar et al., 2014), (b) Contact area calculation (Bansal et al., 2017)

This paper aims to fill the aforementioned knowledge gap. In particular, an ISF power model on a CNC machine is established. At first, the theoretical model of machine tool standby power, feed axis power and sheet forming power is derived based on the contact area and sheet flow condition. Then, the unknown coefficients in the model are fitted on the XKA741/A milling machine. And the developed power prediction model is established. In addition, verification experiments are carried out and the prediction accuracy is evaluated under different process parameters. Then the effects of parameters (step down, sheet thickness, forming tool radius and feed rate) on processing power, processing energy, processing power efficiency and processing energy efficiency are analyzed respectively. Finally, the combination of process parameters that are most conducive to energy conservation is obtained.

2 Methods Due to the complicated wiring method of the machine tool cabinet, the power of each part is difficult to measure separately. However, the power of three parts (𝑃𝑆𝑡𝑎𝑛𝑑, 𝑃𝐹𝑒𝑒𝑑 and 𝑃𝑆ℎ𝑒𝑒𝑡) can be obtained through measuring the total power at different states and then calculating the differences.

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Fig. 3. Machine tool power curves measured at different states

Power measured at different processing states (standby state, air forming state, processing state) on a XKA741/A milling machine is shown in Fig. 3 during ISF. According to the principle of energy consumption decomposition (Zhou et al., 2016), the power of CNC machine tools can be divided into standby power (𝑃𝑆𝑡𝑎𝑛𝑑), feed axis power (𝑃𝐹𝑒𝑒𝑑) and sheet forming power (𝑃𝑆ℎ𝑒𝑒𝑡), as defined by Equation (1). (1)

𝑃𝑀𝑎𝑐ℎ𝑖𝑛𝑒 𝑡𝑜𝑜𝑙 = 𝑃𝑆𝑡𝑎𝑛𝑑 + 𝑃𝐹𝑒𝑒𝑑 + 𝑃𝑆ℎ𝑒𝑒𝑡

As marked in Fig.3, the feed axis power (𝑃𝐹𝑒𝑒𝑑) is the power difference between the air forming power and standby power, while the sheet forming power (𝑃𝑆ℎ𝑒𝑒𝑡) is the difference between processing power and air forming power. Therefore, sheet forming power can be calculated according to Equation (2), since 𝑃𝑆ℎ𝑒𝑒𝑡 equals to zero at the air forming state. (2)

𝑃𝑆ℎ𝑒𝑒𝑡 = 𝑃𝑃𝑟𝑜𝑐𝑒𝑠𝑠 ― 𝑃𝐴𝑖𝑟

Feed axis power can be calculated as shown in Equation (3), since 𝑃𝑆ℎ𝑒𝑒𝑡 equals to zero at the air forming state and both 𝑃𝐹𝑒𝑒𝑑 and 𝑃𝑆ℎ𝑒𝑒𝑡 equal to zero at the standby state. (3)

𝑃𝐹𝑒𝑒𝑑 = 𝑃𝐴𝑖𝑟 ― 𝑃𝑆𝑡𝑎𝑛𝑑

Thus, the energy consumption of the ISF machine tool can be obtained through accumulating the integrals of the various units over time after obtaining the working time of each unit, according to Equation(4). 𝑊𝑀𝑎𝑐ℎ𝑖𝑛𝑒 𝑡𝑜𝑜𝑙 =

∫

𝑡𝑆𝑡𝑎𝑛𝑑

𝑃𝑆𝑡𝑎𝑛𝑑dt + 0

∫

𝑡𝐹𝑒𝑒𝑑

𝑃𝐹𝑒𝑒𝑑𝑑𝑡 + 0

∫

𝑡𝑆ℎ𝑒𝑒𝑡

𝑃𝑆ℎ𝑒𝑒𝑡𝑑𝑡

(4)

0

Therefore, in order to analyze the power of the ISF in detail, the modeling process of the standby power, the feed axis power and the sheet forming power will be provided.

Journal Pre-proof 2.1 Theoretical modelling 2.1.1 Standby power Standby power continuously exists during the whole operating period of the machine tool. In general, standby power can be divided into two types: weak standby power and strong standby power. At the weak standby state, the electric cabinet fan, the display and the control unit works. In addition, machining programming and code loading can be performed, while the NC commands such as feeding and tool setting cannot be executed. At the strong standby state, the high-power circuits (transformer, part of the motor, etc.) of the machine are charged with strong electricity. The machine is in a ready-to-process state and can execute machining commands at any time. The formula to calculate the standby power of the machine tool can be expressed as 𝑃𝑆𝑐𝑟𝑒𝑒𝑛 + 𝑃𝐹𝑎𝑛 + 𝑃𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝐴0 𝑃𝑆𝑐𝑟𝑒𝑒𝑛 + 𝑃𝐹𝑎𝑛 + 𝑃𝐶𝑜𝑛𝑡𝑟𝑜𝑙 + (𝑃𝐿𝑢𝑏 + 𝑃𝑆𝑝𝑐𝑜𝑜𝑙)1 𝐴1 𝑃𝑆𝑡𝑎𝑛𝑑 = 𝑃 = 𝐴 (5) 𝑆𝑐𝑟𝑒𝑒𝑛 + 𝑃𝐹𝑎𝑛 + 𝑃𝐶𝑜𝑛𝑡𝑟𝑜𝑙 + (𝑃𝐿𝑢𝑏 + 𝑃𝑆𝑝𝑐𝑜𝑜𝑙)2 2 𝐴3 𝑃𝑆𝑐𝑟𝑒𝑒𝑛 + 𝑃𝐹𝑎𝑛 + 𝑃𝐶𝑜𝑛𝑡𝑟𝑜𝑙 + (𝑃𝐿𝑢𝑏 + 𝑃𝑆𝑝𝑐𝑜𝑜𝑙)3

{

{

where 𝑃𝑆creen is the power consumed by screen; 𝑃𝐹𝑎𝑛 is the power consumed by electric fan; PControl is the power for controller; 𝑃𝐿𝑢𝑏 is the power consumed by basic lubrication cooling device and 𝑃𝑆𝑝𝑐𝑜𝑜𝑙 is power for spindle forced lubrication. It is noted that powers of the screen, fan and controller are constants. However, 𝑃𝐿𝑢𝑏 changes periodically due to the lubrication cycle but the variation period can be determined by the measurement. When the machine tool started, the frictional damping of the lubrication line is large due to the temperature change of the machine tool, resulting into a large value of 𝑃𝑆𝑝𝑐𝑜𝑜𝑙. While after finishing warming up, 𝑃𝑆𝑝𝑐𝑜𝑜𝑙 tends to be stable and can be regarded as a constant. Specifically, A0 stands for the standby power in weak standby state. A1 stands for the standby power during strong standby state when machine tool starts for a short time and is not warmed up. A2 stands for the standby power when the machine is warmed up and the lubrication works, while A3 stands for the standby power when the machine is warmed up and the lubrication does not work. In this paper, the standby power in strong standby state after warming up is studied and it is defined as a periodically changed value due to the long preparing time before processing and lubrication cycle. 2.1.2 Feed axis power The general feed system of the machine tool consists of servo motors, detection elements, deceleration parts (gear pairs or pulleys), ball screw nut pairs, bearings, moving parts (feed table and slide) and guides. Fig. 4 shows the structure of XKA741/A milling machine feed system which is similar to most of machine tools. As a result, the general models used by other researchers (Altintas et al. 2011), (Zhou et al. 2018) are suitable for XKA741/A milling machine.

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Fig. 4. The structure and composition of XKA741/A milling machine feed system

When the feed system is working, the input power is mainly used to overcome the load torque and friction torque (proportional to the angular velocity). Theoretical formula for the feed axis mechanical demand powers under horizontal and vertical direction are given in Equations (6) and (7) (Altintas et al., 2011), (Zhou et al., 2018)： ℎ𝑝𝜇𝑔𝑓(𝑚𝑡 + 𝑚𝑤)𝑔 + 𝜋𝜇𝑏𝑑𝑏𝐹𝑝 4𝜋2𝑟𝑔2𝐵 𝑃𝑓𝑑𝐻 = 𝑣𝑓 + 𝑣𝑓2 60000ℎ𝑝 360ℎ𝑝2

(6)

𝜋𝜇𝑏𝑑𝑏𝐹𝑝 ± 𝑚𝑡𝑔ℎ𝑝 4𝜋2𝑟𝑔2𝐵 𝑣2 𝑃𝑓𝑑𝑉 = 𝑣𝑓 + 2 𝑓 60000ℎ𝑝 3600ℎ𝑝

(7)

where 𝑟𝑔 stands for the gear reduction ratio; 𝐵 stands for the equivalent viscous damping friction coefficient; ℎ𝑝 stands for ball screw pitch; 𝜇𝑏 stands for friction coefficient of the bearing; 𝑚 stands for the mass of the machined workpiece; 𝑔 stands for the gravitational acceleration and 𝑑𝑏 stands for the average diameter of the supporting bearing. Since the algebraic coefficient before 𝑣𝑓 or 𝑣𝑓2 in Equations (6) involves system design parameters that are difficult to obtain, it can be represented by a coefficient C which can be obtained by regression fitting with experimental results. Therefore, the empirical model (Kim et al., 2015) for obtaining the feed power can be given by: 𝑃feed(𝑣𝑓) =

{

𝑃𝑓𝑑𝐻 = 𝐶1 + 𝐶2v𝑓 + 𝐶3𝑣𝑓2 𝑃𝑓𝑑𝑉 = 𝐶'1 + 𝐶'2v𝑓 + 𝐶'3𝑣𝑓2

(8)

It should be noted that the feed direction includes horizontal feed and vertical feed in general. As for Z axis, the feed direction includes up and down in horizontal feed. While as for X and Y feed axis, the feed direction only includes horizontal feed. Coefficients 𝐶1~𝐶3，𝐶'1~𝐶'3 differ depending on the feed direction and feed axis. As a result, the unknown coefficients need to be determined through four groups of

Journal Pre-proof experiments. 2.1.3 Sheet forming power Sheet forming power is the part of the power used directly to deform the sheet only. In the case of the same material type and wall angle, the sheet forming power is related to the processing parameters and can be expressed by the following formula: 𝑃𝑆ℎ𝑒𝑒𝑡 = 𝑓(𝑣𝑓,𝑡0,𝑟,∆𝑧)

(9)

where 𝑣𝑓 stands for feed rate; 𝑡0 stands for initial sheet thickness; 𝑟 stands for forming tool radius and ∆𝑧 stands for step down. According to the conservation law of energy and recent literatures of cutting process (Avram and Xirouchakis, 2011), (Yoon et al., 2013), (Wang et al., 2013), (Shao et al., 2004), the sheet forming power can be obtained by calculating the mechanical power through force and speed. In this section, a theoretical prediction model for sheet forming power is established based on the contact condition between the sheet and the forming tool as well as the sheet flow behavior during the ISF process. The actual contact characteristic during the forming process is carefully observed as shown in Fig. 5.

Fig. 5. Indentation on the surface of the sheet and worn forming tool (a) Indentation on the surface of the sheet (AA-2024, d=10mm, t0=1.5mm, 𝛼 = 55。), (b) Worn forming tool (M3 high speed steel)

Fig. 5(a) shows the indentation on the sheet induced by the forming tool after stopping the feed axis during certain pass and this indentation reflects the contact condition between the forming tool and the sheet. It can be observed that the contact area is a spatial concave surface constructed by the spherical surface of the forming tool, the cylinder of the previous pass and the upper surface of the unformed sheet. Meanwhile, points A, B and C are defined at the border of the contact area, as shown in Fig. 5(a). Fig. 5(b) shows the worn forming tool (M3 high-speed steel) after deforming a pyramid (AA-2024 with a thickness of 2 mm). The sheet is prepared with a Ni-Cr coating by atmospheric plasma spraying to accelerate the tool wear. It can be observed from the image that the forming tool is severely worn, resulting in four L-shaped pits witch are centrally symmetrical. Within the pits, the position away from the tool center

Journal Pre-proof is deeper. The scratch is generated by the relative direction of motion between the forming tool and sheet. And the points at both ends of the pits are defined as points B' and C'. These two points correspond to points B and C on the sheet surface during the actual machining process.

Fig. 6. Scanned curves of at the bottom corner of the deformed pyramids under different process parameters (measured by a D-RAPID 3D laser measuring machine) (a) Scanned curves at difference sheet thicknesses, (b) Scanned curves at different forming tool diameters, (c) Enlarged figure of the mark of ℎ1 in figure(a) and (b), (d) Definition of the contact angle.

The variation trend of the actual contact angle is further analyzed through experimental measurements. Fig. 6(a) and (b) shows the scanned curves at the bottom corner of the deformed pyramids with a wall angle of 55° at different sheet thicknesses and tool diameters. Define the contact angle 𝛽 as the center angle of the corner at the bottom edge when performing a pass as shown in Fig. 6(d), which is related to the angle 𝛼 and the distance ℎ1. Since the angle 𝛼 is equal to the wall angle which is constant in this research, the distance ℎ1 from the lowest point of the deformed sheet surface to the bottom horizontal plane can reflect the size of the contact angle. As a result, as shown in Fig. 6(c), it can be inferred that the contact angle 𝛽 increases significantly as the sheet thickness increases. However, angle 𝛽 increases significantly when the tool diameter changes from 10 mm to 15 mm, while the change of 𝛽 is not obvious when the tool diameter increases from 15 mm to 20 mm.

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Fig. 7. Contact area model (a) 3D model during the ISF process with contact area, (b) Contact area model plotted in MATLAB.

According to the observed characteristics, it can be considered that the contact area between the forming tool and the plate is a spatial surface: a quarter-spherical surface which is cut by a cylindrical surface and the upper surface of the unformed sheet, as shown in Fig. 7. Fig. 7(a) shows the 3D model during the ISF process, where the red area indicates the contact area. Fig. 7(b) shows the contact area plotted in a defined coordinate system, where the selected quarter sphere is located in the 5th and 6th octant of the coordinate system. Based on above observations, the following assumptions are proposed: 1) Around the contact area, the upper surface of the unformed sheet is a sloping plane. 2) The sheet thickness variation is ignored along the feed direction, and the thinning along the side wall direction is uniform. 3) Assume that angle β changes linearly with the thickness of the sheet and the radius of forming tool within a certain range while infinitely close to two fixed values at both ends. According to the assumption, a contact area schematic diagram is established, as shown in Fig. 8. Fig. 8(a) and (b) shows the contact condition in main view and left view. Fig. 8(c) shows the sheet flow condition in 𝛾 position and Fig. 8(d) shows the isometric drawing. It must be pointed out that the contact area model and assumptions are suitable for single-point incremental sheet forming with large deformation of thin sheets. In addition, if the rotary shape (e.g. cone) is processed, the cylindrical surface in the contact area of the pyramid should be replaced by a circular cylindrical surface.

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Fig. 8. Contact area model of ISF for calculation (a) Main view, (b) Left view, (c) Movement of material in 𝛾 plate, (c) the isometric drawing.

During the ISF process, the deformation of the sheet is essentially produced by the forming force between the forming tool and the sheet which can be broken down into tangential force and normal force. The tangential force causes frictional power, and the normal force directly causes sheet deformation. As a result, the sheet forming power 𝑃𝑆ℎ𝑒𝑒𝑡 can be divided into deformation-induced power and friction-induced power and can be solved separately as: (10)

𝑃𝑆ℎ𝑒𝑒𝑡 = 𝑃𝐷𝑒𝑓𝑜𝑟𝑚 + 𝑃𝐹𝑟𝑖𝑐𝑡 (1) Deformation-induced power The equation for the spherical surface of the forming tool can be express as 2

2

2

()

𝑑 𝑥 +𝑦 +𝑧 = 2 2

(11)

𝑧 ≤ 0,y ≥ 0

The cylindrical surface equation formed by the previous pass can be expressed as 2

(

(𝑧 ― ∆𝑧) + 𝑥 +

∆𝑧

2

𝑡𝑎𝑛𝛼

)

2

()

𝑑 = 2

𝑧≤0

(12)

As shown in Fig. 8(c), when the forming tool squeeze the sheet, the friction is too small that it can be considered that the force of the forming tool against the sheet is always perpendicular to the spherical surface. When the sheet is squeezed and

Journal Pre-proof deformed by the spherical forming tool, the motion track of a certain point on the sheet can be considered as a part of the cycloid curve of which the length is 𝑙. The equation of cycloid curve is expressed as 𝑟𝑠𝑖𝑛𝜑 {𝑥𝑦==𝑟𝜑𝑟 ――𝑟𝑐𝑜𝑠𝜑

(13)

𝑙 can be derived as 𝑙(𝛾) =

∫

2𝜋

2 2 𝑥' (𝜑) + 𝑦' (𝜑)𝑑𝜑

(14)

2𝜋 ― 𝜔

The lower limit of integral (2𝜋 ― 𝜔) changes with 𝛾. As shown in Fig. 8(b), 𝜔 can be derived by dr and 𝑟, defined as 𝑟 ― 𝑑𝑟(𝛾) 𝜔(𝛾) = 𝑎𝑟𝑐𝑐𝑜𝑠 (15) 𝑟

(

)

According to Fig. 8(b), dr(𝛾) can be calculated by the following formula based on geometric relation. 𝑑𝑟(𝛾) = 𝑟

) (

(

∆𝑧 ― 2 𝑐𝑜𝑠 (𝛾) + 2∆𝑧𝑠𝑖𝑛(𝛾) + 𝑡𝑎𝑛𝛼

∆𝑧 2 𝑐𝑜𝑠 (𝛾) + 2∆𝑧𝑠𝑖𝑛(𝛾) 𝑡𝑎𝑛𝛼

―

)

2

(( )

―4

∆𝑧 𝑡𝑎𝑛𝛼

2

)

+ ∆𝑧2 ― 𝑟2

2

(16) The energy required to actually deform a micro element on the sheet can be expressed as (17)

𝑑𝑊 = 𝜎𝑡(𝛾) × 𝑙(𝛾)

where 𝜎t and 𝑙 are functions related to 𝛾. The value of 𝑙 is given by Equation (17). 𝜎t can be calculated by Membrane analysis (Chang et al., 2019). 2 2𝑡 𝜎𝑡 = ― 𝜎 3𝑟 + 2.5𝑡 The thickness variation of the sheet can be expressed by the following formula. 𝑡0(1 ― 𝑐𝑜𝑠𝛼)𝛾 (18) 𝑡 = 𝑡0 ― 𝛽 The calculation of 𝜎t equation can be expressed as 2 𝜎𝑡 = ― 𝜎 3

𝑡0(1 ― 𝑐𝑜𝑠𝛼)𝛾

(

2 𝑡0 ―

(

𝛽

(

𝑟𝑡𝑜𝑜𝑙 + 2.5 𝑡0 ―

)

𝑡0(1 ― 𝑐𝑜𝑠𝛼)𝛾 𝛽

))

(19)

Due to work hardening occurred at the sheet, σ should be substituted by σ = 𝜎0 𝑝 + 𝐾(1 ― 𝑒𝑛𝜀 ) (Chang et al., 2019). Therefore, deformation-induced power can be obtained 𝛽

𝑃𝐷𝑒𝑓𝑜𝑟𝑚 = 𝑣𝑓 ×

∫ 𝜎 (𝛾) × 𝑙(𝛾)𝑑𝛾 0

t

(20)

Journal Pre-proof (2) Friction-induced power As shown in Fig. 8(c), at certain position on contact line in XOZ plane, the moving distance of the forming tool can be expressed as 𝐿' = 𝑡 × 𝑣𝑓 × 𝑟 × 𝑠𝑖𝑛 (𝜔)

(21)

However, due to the bending deformation of the sheet, the length of friction between the forming tool and the sheet can be calculated as follows (22)

𝐿 = 𝜔(𝛾) × 𝑟 The shear stress at each point can be expressed as

(23)

𝜏𝜃 = 𝜎t(𝛾) × 𝜇𝜃

The friction-induced power can be obtained by integral in the contact range as derived in Equation (24). 𝛽

𝑃𝐹𝑟𝑖𝑐𝑡 =

∫ 𝜎 (𝛾)𝜇𝑟𝜔(𝛾)𝑑𝛾

(24)

𝑡

0

As a result, the sheet forming power can be expressed by 𝑃𝑆ℎ𝑒𝑒𝑡 = 𝑣𝑓 ×

∫

𝛽 0

𝜎t(𝛾) × 𝑙(𝛾)𝑑𝛾 + 𝑣𝑓 ×

𝛽

∫ 𝜎 (𝛾)𝜇𝑟𝜔(𝛾)𝑑𝛾 𝑡

(25)

0

Within the developed model, the contact angle 𝛽 and coefficient of friction should be determined through forming experiments.

2.2 Experiment 2.2.1 Experimental equipment In this study, the XKA714/A CNC milling machine, as shown in Fig. 9(a), is used for forming tests and CW240 power meter, as shown in Fig. 9(c), is used for measuring the power during experiments. Electric power consumption data is analyzed by CW Viewer AP240E and MATLAB. The experimental setup is shown in Fig. 9. The CNC milling machine features with high load carrying capacity and a fast feed rate up to 8000 mm/min. And Table 1 shows the typical performance parameters of the XKA714/A CNC milling machine. During the experiment, the power meter is connected to the main line of the machine tool power distribution box as shown in Fig. 9(b).

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Fig. 9. Experimental equipment (a) XKA714/A milling machine, (b) Wiring position, (c) CW240 power meter. Table 1 XKA714/A machine tool performance parameters Performance parameters

Size

Workbench area/mm (width × length)

400*1100

X-axis stroke length/mm (left and right)

600

Y-axis stroke length/mm (front and rear)

450

Z-axis stroke length/mm (up and down)

500

Feed rate (mm/min)

X,Y: 6-3200; Z:3-1600

Fast moving feed rate (mm/min)

X,Y:8000;Z:4000

Lubrication power and cycle

3 W/15 min

2.2.2 Experimental design 1) Experimental design for standby power model In this section, the method to measure the standby power is introduced. At first, power on the electric cabinet and energize the control system. The lubrication and cooling device then starts to work. After a warming up for about 10 minutes, the strong standby power is recorded by the power meter for another 5 minutes. 2) Experimental design for feed axis power model The test method is as follows: X, Y and Z axis of the machine tool are moved back and forward in the horizontal or vertical direction at the feed speed v𝑓 and the moving distance is determined according to the stroke of the machine tool as shown in Table 1. The power data during the entire feeding process is collected by power meter. Then, the feed axis power is the difference between the collected machine tool power and the standby power. According to the performance of the milling machine in Table 1, set the feed

Journal Pre-proof speed of the machine to a range of 200~3600 mm/min. For each 200 mm/min increase, move the X and Y axes back and forward in the horizontal direction. The moving distance is between -150 mm and +150 mm. The Z axis moves up and down in the vertical direction with a relative distance of 0 mm to -300 mm. 3) Experimental design for sheet forming power model In this experiment, the sheet forming power is obtained by measuring the difference between processing power and air forming power within the same depth. After processing the workpiece, an additional air forming experiment with the same tool path needs to be performed immediately. Then the processing time is calculated according to processing parameters and depth. Four sets of fitting tests are designed after selecting step down, forming tool radius, feed rate and sheet thickness as experimental variables. A pyramid with a forming angle of 55°, a depth of 20 mm and an opening length of 100 mm was selected as the target shape. The experimental design is shown in Table 3. Once fracture occurred, the forming process is stopped. Since the sheet forming power is constantly changing with the change of depth, the average forming power within a depth of 9.6mm is evaluated. Table 2 shows the experimental design. Table 2 Experimental design for sheet forming power model Test number

Step down

Forming tool

Thickness (mm)

Feed rate (mm/min)

(mm)

(radius /mm)

1

0.8

5

1.5

2000

2

0.4

5

1.5

2000

3

1.2

5

1.5

2000

4

0.8

7.5

1.5

2000

5

0.8

10

1.5

2000

6

0.8

5

1

2000

7

0.8

5

2

2000

8

0.8

5

1.5

1000

9

0.8

5

1.5

3000

4) Experimental design for model verification Verification experiment is carried out to verify the prediction accuracy of the model. The pyramid with the same dimension is used and the experiment is designed by orthogonal experimental design method with four factors and three levels. The experimental design is shown in Table 3. Table 3 Experimental design for model verification Test number

Step down (mm)

Forming tool

Thickness (mm)

radius (mm)

Feed rate (mm/min)

V1

0.4

10

1

1000

V2

0.8

10

1.5

2000

V3

1.2

10

2

3000

Journal Pre-proof V4

0.8

5

2

1000

V5

1.2

5

1

2000

V6

0.4

5

1.5

3000

V7

1.2

7.5

1.5

1000

V8

0.4

7.5

2

2000

V9

0.8

7.5

1

3000

3 Model development and verification 3.1 Development of the standby power model The measured power curve and the smoothed curve at the strong standby state are shown in Fig. 10. Although the standby power has small fluctuations, the overall trend tends to be stable. After averaging the measured data, a standby power of 0.57556 kW (𝐴2 in Equation (5)) is obtained for the machine.

Fig. 10. Measured machine tool power curve under standby state

3.2 Development of feed axis power models Table 4 Feed axis power measured in X-direction 𝑣𝑓(mm/min)

200

400

600

800

1000

1200

1400

1600

1800

𝑃𝑓𝑑𝑥(kW)

0.0085

0.0151

0.0183

0.0219

0.0257

0.0296

0.0360

0.0424

0.0467

𝑣𝑓(mm/min)

2000

2200

2400

2600

2800

3000

3200

3400

3600

𝑃𝑓𝑑𝑥(kW)

0.0559

0.0640

0.0744

0.0757

0.0856

0.0947

0.1043

0.1047

0.1134

Table 5 Feed axis power measured in Y-direction 𝑣𝑓(mm/min)

200

400

600

800

1000

1200

1400

1600

1800

𝑃𝑓𝑑𝑦(kW)

0.0109

0.0178

0.0230

0.0299

0.0363

0.0437

0.0509

0.0599

0.0707

𝑣𝑓(mm/min)

2000

2200

2400

2600

2800

3000

3200

3400

3600

𝑃𝑓𝑑𝑦(kW)

0.0809

0.0864

0.1013

0.1090

0.1164

0.1275

0.1363

0.1492

0.1623

Table 6

Journal Pre-proof Feed axis power measured in Z-up-direction 𝑣𝑓(mm/min)

200

400

600

800

1000

1200

1400

1600

1800

𝑃𝑓𝑑𝑧𝑢(kW)

0.0202

0.0408

0.0652

0.0918

0.1173

0.1444

0.1705

0.2069

0.2575

𝑣𝑓(mm/min)

2000

2200

2400

2600

2800

3000

3200

3400

3600

𝑃𝑓𝑑𝑧𝑢(kW)

0.2786

0.3136

0.3419

0.3594

0.3706

0.3711

0.3803

0.4389

0.4553

Table 7 Feed axis power measured in Z-down-direction 𝑣𝑓(mm/min)

200

400

600

800

1000

1200

1400

1600

1800

𝑃𝑓𝑑𝑧𝑑(kW)

-0.0111

-0.0228

-0.0316

-0.0341

-0.0391

-0.0437

-0.0524

-0.0513

-0.0572

𝑣𝑓(mm/min)

2000

2200

2400

2600

2800

3000

3200

3400

3600

𝑃𝑓𝑑𝑧 𝑑(kW)

-0.0505

-0.0650

-0.0741

-0.0780

-0.0773

-0.0528

-0.0814

-0.0754

-0.0833

Table 4 to Table 7 present the measured values for the feed axis power at different feed rates. Then, the measured feed axis power is fitted by MATLAB to determine the parameters of the feed axis power model. As a result, the X-axis feed axis power model (R-Square=99.46%) can be expressed as 𝑃𝑓𝑑𝑥 = 3.33 × 10 ―9 × 𝑣𝑓2 + 1.92 × 10 ―5 × 𝑣𝑓 + 4.391 × 10 ―3

(26)

Y-axis feed axis power model (R-Square=99.88%) can be expressed as 𝑃𝑓𝑑𝑦 = 4.025 × 10 ―9 × 𝑣𝑓2 + 2.915 × 10 ―5 × 𝑣𝑓 + 4.211 × 10 ―3

(27)

Z-up-axis feed axis power model (R-Square=98.88%) can be expressed as 𝑃𝑓𝑑zu = ―1.055 × 10 -8 × v𝑓2 + 1.718 × 10 ―4 × 𝑣𝑓 ― 3.112 × 10 ―2

(28)

Z-down-axis feed axis power model (R-Square=96.43%) can be expressed as 𝑃𝑓𝑑zd = 3.968 × 10 -9 × v𝑓2 ± 3.525 × 10 ―5 × 𝑣𝑓 ― 7.713 × 10 ―3

(29)

The correlation coefficients are greater than 95%, which means that there is a significant correlation between the feed power and the feed rate.

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Fig. 11. Fitted curves and measured points for feed axis power (a)Feed axis power of X axis, (b) Feed axis power of Y axis, (c) Feed axis power of Z axis in down direction, (c) Feed axis power of Z axis in up direction

The fitted curves for the feed axis power are illustrated in Fig. 11. It is evident that the feed power for both X and Y axes increase as increasing the feed rate. Under the same feed rate, the X-axis feed power is less than the Y-axis though the X and Y axes use the same type of 1FT6074 AC servo motor. This phenomenon is caused by the difference in friction between the X-axis and Y-axis. When the machine moves along the Y-axis, the friction is greater than the X-axis direction. However, this phenomenon provides convenience for analyzing the power analysis when processing a pyramid in ISF. As shown in Fig. 11(c), the power of the Z-up-axis is significantly larger than that in other directions since the headstock moves upward to overcome gravity and greater frictional resistance. The power of the Z-down-axis is small and negative, since the gravity contributes positive work and some sdandby devices turns off.

3.3 Development of the sheet forming power model Suppose the minimum value of 𝛽 is 65° according to the observation in Fig. 6. Then the unknown parameters (contact angle 𝛽 and friction coefficient 𝜇𝜃) within the boundary in the sheet forming power model are obtained in MATLAB through optimum seeking method. Within the boundary condition (β varies from 65° to 125°, 𝜇𝜃 varies from 0.01 to 0.20), the parameters to be solved are increased incrementally. The sheet forming power and relative error are calculated by substituting different sizes of parameters. The values corresponding to the sheet forming power with the smallest average relative error is selected as the optimal solution of the unknown

Journal Pre-proof parameters. The results show that when the value of β is 82.3o in Test 1 (sheet thickness is 1.5 mm and forming tool radius is 5 mm) and friction coefficient μθ is 0.20, the model has the best prediction accuracy. The average error is within 10% Thus the calculation expression of β is as shown in Equation (30) where the value 0.2369 is obtained through substituting 65° into the last formula in Equation (30).

𝛽=

{

𝑡0

65°

𝑟

< 0.2369

82.3° × 5 × 𝑡0 1.5 × 𝑟

0.2369 ≤

𝑡0

(30)

𝑟

Fig. 12 shows the comparison of the predicted sheet forming power curve and measured sheet forming power points. And it can be seen from the figure that the prediction curve of sheet forming power agrees well with the measured points.

Fig. 12. Predicted sheet forming power curve and measured power points (a) Effect of forming tool, (b) Effect of feed rate, (c) Effect of step down, (d) Effect of sheet thickness.

Admittedly, the proposed sheet forming power can not be directly applied to any cases since the unknown coefficients should differ when the experimental condition changes. However, the basic sheet forming power model and the development method are universal and can be adopted. Specifically, if the tool diameter and sheet thickness are unchanged, only the friction coefficient and the contact angle need to be determined through experiments.

3.4 Verification of the developed models Before calculating and comparing, it should be noted that all feed axes work

Journal Pre-proof during ISF process. Therefore, to calculate the predicted processing power, the feed axis power 𝑃𝐹𝑒𝑒𝑑 is the average value in X and Y direction, since the feed axis power in Z direction is neglected due to its small value and a short running time. Additional experimental tests are performed to verify the reliability of the developed power models and Fig. 13 shows the predicted power and absolute error compared with measured values. It can be observed from the figure that the prediction model has good accuracy. The absolute error is within 10 W in the four prediction models. Meanwhile the results show that the standby power model and processing power model have excellent prediction accuracy and the relative error is within 5% as shown in Fig. 13(a) and (d).The relative error of feed axis power is within 10%, as shown in Fig. 13(b). In addition the relative error of sheet forming power is within 15% except for test V8 of which the relative error is within 20% as shown in Fig. 13(c).

Fig. 13. Predicted power and absolute error of verification experiment (a) Standby power and absolute error, (b) Feed axis power and absolute error, (c) Sheet forming power and absolute error, (d) Processing power and absolute error.

Prediction errors are analyzed and may be caused by the following reasons: The fluctuation of the machine tool power. The power of several operating units of the machine tool such as lubrication system and cooling system may be influenced by changes of both the environmental temperature and the working temperature of machine tool. The clamping condition. When the sheet clamped, the sheet may initially bend slightly which influences the determination of processing depth. As a result, the sheet forming energy may be influenced. The lubrication condition between forming tool and sheet. The change of the quantity and uniformity of lubricating oil may influence the recording of sheet forming

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4 Results and discussions 4.1 Prediction of processing power and power efficiency Since the sheet forming power and the feed axis power directly cause deformation of the sheet, the sum of the sheet forming power and feed axis power is defined as the useful power, denoted by PUse. Most of the standby power is used for heat dissipation, hydraulic equipment, cooling equipment, etc., while these devices are not necessary for deforming the material sheet. Therefore, the ratio of useful power to processing power is defined as the efficiency of the machine tool in ISF, denoted by 𝜓, given by: 𝜓=

PUse PProcess

=

PMaterial + PFeed PProcess

(31)

The power efficiency indicator reflects the energy saving capabilities of the machine tool and it provides guidance on the design and selection of energy-efficient forming machines. According to the prediction model, the response surface of 𝑃𝑃𝑟𝑜𝑐𝑒𝑠𝑠 and 𝜓 within a 9.6mm depth can be obtained as shown in Fig. 14 and Fig. 15. The process parameters are set as forming tool radius (5 mm), feed rate (2000 mm/min), step down (0.8 mm) and sheet thickness (1.5 mm), if no otherwise specified in the figures.

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Fig. 14. Effects of four process parameters on processing power and power efficiency surface (a) Influence of forming tool radius and feed rate on processing power, (b) Influence of step down and sheet thickness on processing power, (c) Influence of forming tool radius and feed rate on power efficiency, (d) Influence of step down and sheet thickness on power efficiency.

Fig. 14 shows the effects of four process parameters on the processing power and power efficiency. The effect of feed rate on the processing power and power efficiency is most pronounced, as shown in Fig. 14(a) and (c). The sheet thickness and step down have a similar effect on the processing power and power efficiency, as shown in Fig. 14(b) and (d), while less than feed rate. In addition, the effect of the forming tool radius is unnoticeable, as shown in Fig. 14(a) and (c). The detailed influencing mechanism of each parameter on processing power and power efficiency are discussed below.

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Fig. 15. Predicted processing power and power efficiency influenced by process parameters (a) Effect of forming tool radius on processing power and power efficiency, (b) Effect of feed rate on processing power and power efficiency, (c) Effect of step down on processing power and power efficiency, (d) Effect of sheet thickness on processing power and power efficiency.

1) Influence of forming tool radius Fig. 15(a) shows the variation trend of the processing power and power efficiency at different tool sizes. It can be seen that when the tool radius increases, the sheet forming energy consumption reduces at first and then increases which is similar to published experimental work (Y. Li et al., 2015). According to the above analysis, the forming tool radius mainly affects processing power and power efficiency through sheet forming power as shown in Equations (20) and (24) since the change of radius causes the change of contact area and angle. Nevertheless, the effect of radius on processing power and power efficiency is not obvious, since sheet forming power only occupies a small proportion of processing power. 2) Influence of feed rate Fig. 15(b) shows the effect of feed rate on processing power and power efficiency. It can be observed that when feed rate changes from 1000 mm/min to 4000 mm/min, the processing power is increased for 150 W and power efficiency is increased by 20%. The reason is that both the feed axis power and sheet forming power are increased with increasing the feed rate, according to Equations (20), (24), (27) and (28). In addition, the influence of the feed rate on the sheet forming power is mainly reflected by the change of the length of the formed sheet per unit time in the feed direction. As a result, the power efficiency can be also improved with the increase of feed rate.

Journal Pre-proof 3) Influence of step down Fig. 15(c) shows the variation trend of processing power and power efficiency at different step downs. It can be seen that when step down changes from 0.5 mm to 1.5 mm, the processing power is increased about 20 W and the power efficiency is increased by 2.5%. The step down has less influence on both processing power and power efficiency, since it mainly affects processing power and power efficiency through sheet forming power which is similar to the effect of forming tool radius. The increase of the step down results in an increase in the length of the motion track 𝑙 and the friction length 𝐿 at each point in the forming region, as described in Equations (14) and (22). This leads to an increase of the sheet forming power and also the power efficiency. 4) Influence of thickness Fig. 15(d) shows the influence of sheet thickness on the processing power and power efficiency. It can be seen that when thickness changes from 0.5 mm to 15.mm, the processing power increases about 15 W and power efficiency increases about 2.5%. The sheet thickness has less influence on both processing power and power efficiency. The reason is that the thickness mainly affects processing power and power efficiency through sheet forming power, which is similar to the influence of forming tool radius and step down. The increase of sheet thickness leads to the increase of strain in thickness direction. Meanwhile the contact area is also increased by affecting the contact angle. Therefore, the increase of sheet thickness causes the increase of processing power and power efficiency.

4.2 Prediction of processing energy and energy efficiency Similarly, the sum of the sheet forming energy and the feed axis energy is defined as useful energy, denoted by 𝐸𝑈𝑠𝑒. The ratio of useful energy to processing energy is defined as the energy efficiency of the machine tool in ISF, denoted by 𝜙, given as: 𝐸𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙 + 𝐸𝐹𝑒𝑒𝑑 EUse (32) 𝜙= = 𝐸𝑃𝑟𝑜𝑐𝑒𝑠𝑠 𝐸𝑃𝑟𝑜𝑐𝑒𝑠𝑠 The combination of parameters with high energy efficiency saves more energy when using the same machine tool. Therefore, the energy efficiency indicator reflects the energy saving capabilities of the combination of parameters. According to the prediction model, the response surfaces and variation of 𝐸𝑃𝑟𝑜𝑐𝑒𝑠𝑠 and 𝜙 within 9.6mm depth are obtained as shown in Fig. 16 and Fig. 17. Predicted processing energy and energy efficiency. The process parameters are set as forming tool radius (5 mm), feed rate (2000 mm/min), step down (0.8 mm) and sheet thickness (1.5 mm), if no otherwise specified in the figures

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Fig. 16. Effect of two different process parameters on processing energy and energy efficiency surface (a) Influence of forming tool radius and feed rate on processing energy, (b) Influence of step down and sheet thickness on processing energy, (c) Influence of forming tool radius and feed rate on energy efficiency, (d) Influence of step down and sheet thickness on energy efficiency.

Fig. 16 shows the effects of four process parameters on processing energy and energy efficiency surface. As can be seen, the influence of feed rate and step down on processing energy consumption is significant as shown in Fig. 16(a) and (b), while forming tool radius and sheet thickness have less effect on processing energy. Moreover, in terms of the energy efficiency as shown in Fig. 16(c) and (d), the feed rate has the greatest impact on energy efficiency, followed by the step down and sheet thickness, and finally the forming tool radius. The detailed influencing mechanism of each parameter is discussed below.

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Fig. 17. Predicted processing energy and energy efficiency influenced by process parameters (a) Effect of forming tool radius on processing energy and energy efficiency, (b) Effect of feed rate on processing energy and energy efficiency, (c) Effect of step down on processing energy and energy efficiency, (d) Effect of sheet thickness on processing energy and energy efficiency.

1) Influence of forming tool radius As can be seen from Fig. 17(a), both processing energy and energy efficiency decrease first and then increase as the radius of the forming tool increases. This is because that the contact angle is decreased first and then fixed at a certain value (65° in this case ) due to the contact limit as assumed in 3.3. And the forming tool radius has no influence on forming time. As a result, the trends of both processing energy and energy are similar. When the forming tool radius is changed from 5mm to 6.5mm, the processing energy is reduced by about 700 J, by approximately 0.45%. The energy efficiency is reduced by approximately 0.5%. Nevertheless, since the influence of the forming tool radius on the processing energy is mainly reflected on the sheet forming energy, which takes only a small proportion of the processing energy, the forming tool size has limited effect on both processing energy consumption and energy efficiency. 2) Influence of feed rate Fig. 17(b) shows the variation trend of processing energy and energy efficiency at different feed rates. It can be observed that the total energy of the machine decreases drastically with the feed rate increasing within the range of 400 mm/min. As the feed rate increases between 1000 mm/min and 4000 mm/min, the decrease rate of processing energy tends to be slow. However, in terms of the energy efficiency, it increases linearly as the feed rate increases over the whole range. In particular, the efficiency is greatly improved from 7% to 25% when the feed rate increases from 1000

Journal Pre-proof mm/min to 4000 mm/min. Increasing the feed rate not only reduces the processing energy, but also increases the energy efficiency. Therefore, for ISF, increasing the feed rate is a very energy-saving strategy. The reason is that the effect of feed rate on processing energy consumption mainly reflects on processing time as shown in Fig. 18 (a). In ISF process, the standby power occupies a considerable proportion of the processing power. When the feed rate is 2000 mm/min, the standby power accounts for more than 85% of the processing power, as shown in Fig. 18(c), and the feed power accounts for 10.17%, sheet forming power 2.11%. When the feed rate increases, although the sheet forming power and feed axis power increase, the processing time is greatly reduced, as shown in Fig. 18 (a). According to Equations (20) and (24), the sheet forming energy consumption is unchanged and the feed axis power only increases slightly. However, reduced processing time decreases total standby time, resulting in the reduction of standby energy. The decreased standby energy is much larger than the increased feed energy. As a result, the processing energy consumption greatly reduces and the energy efficiency increases.

Fig. 18. Power composition and processing time (a) Processing time influenced by feed rate (step down 0.8 mm), (b) Processing time influenced by step down (feed rate 2000 mm/min), (c) Power composition of Test V5 in section 2.2.2.

3) Influence of step down Fig. 17(c) shows the influence of step down on processing energy consumption and energy efficiency. Similar to the feed rate, the processing energy consumption of the process decreases sharply as increasing step down within a small range. Then, the processing energy tends to decrease slowly with further increasing the step down. The efficiency gradually increases as the step down increases. Specifically, when the step size increases from 0.2 mm to 1.5 mm, the processing energy is decreased by 4.5×105 J and the energy efficiency is increased from 12% to 15%. Therefore, increasing step down is also an effective energy-saving method for ISF. Similarly, the main effect of the step down on processing energy consumption is

Journal Pre-proof also reflected on the processing time as shown in Fig. 18 (b). As mentioned above, standby power occupies a considerable proportion of the total processing power as shown in Fig. 18 (c). Increasing step down also reduces the processing time, resulting in effective reduction of standby energy consumption. Although the sheet forming power increases as the step down increases, the sheet forming energy consumption changes slightly due to the reduction in processing time. In addition, the increase of step down increases the energy efficiency by the way of increasing sheet forming power. 4) Influence of thickness Fig. 17(d) presents that both processing energy consumption and energy efficiency increase as the sheet thickness increases. When the thickness changes from 0.5 mm to 1.5 mm, the processing energy is increased by 2.29% and the energy efficiency is increased approximately 2%. Since the influence of sheet thickness on the processing energy is mainly reflected on the sheet forming power, it only affects the sheet forming energy consumption. Therefore, the thickness variation only has limited effect on both processing energy consumption and energy efficiency. In summary, the best method to decrease energy consumption is to select a machine tool with larger power efficiency and a combination of parameters with larger energy efficiency. Therefore, it is effective for energy saving to increase the feed rate and step down, to decrease the sheet thickness, while select a proper size of the forming tool (depending on the sheet material and wall angle). As for the case studied in this paper, the most energy efficient process parameter combination is obtained, with 4000 mm/min feed rate, 1.2 mm step down, 7.5 mm forming tool radius and 1 mm sheet thickness.

5 Conclusions ISF is a new and green manufacture process for small batch production. In order to analyze its environmental performance, the electric power and energy consumption of ISF process are studied comprehensively. An energy consumption model of ISF is firstly proposed which combines the idea of machine tool energy decomposition and the contact area determination in ISF. Moreover, the model proposed in this paper can not only predict the energy consumption but also explain the physical phenomenon through the control of variables in the model. The following important conclusions can be drawn: 1) Theoretical power models for ISF process are proposed, including the standby power model, feed axis power model and sheet forming power model. In particular, the theoretical mechanism model of sheet forming power is established according to the observed contact condition between sheet and forming tool as well as the sheet flow condition. 2) The unknown coefficients in theoretical models of standby power, feed axis power and sheet forming power are determined based on corresponding experiments. Further

Journal Pre-proof verification experiments are carried out and results show that the error of processing power prediction model is within 5%. 3) Based on the established theoretical models, the effects of four process parameters on processing power, power efficiency, processing energy and energy efficiency are analyzed. The results show that the feed rate has the greatest impact, since it influences the sheet forming power, feed axis power and processing time. The step down has a great influence on energy and energy efficiency, while has little effect on processing power and power efficiency, since the step size affects the sheet forming power and processing time. The influences of the tool radius and the sheet thickness on processing power, power efficiency, processing energy and energy efficiency are not obvious since the both can only affect the sheet forming power. 4) An in-depth analysis of the energy consumption in ISF is provided. The best combination of process parameters is obtained, which is most beneficial to save energy in this case, with 4000 mm/min feed rate, 1.2 mm step down, 7.5 mm forming tool radius and 1 mm sheet thickness. Since the fracture limits the study of sheet forming power in a certain depth, the effect of depth on contact angle and sheet forming power will be studied in the future work. In addition, improvement of the forming equipment is another effective approach to save energy for ISF process.

Acknowledgements: This work is supported by National Natural Science Foundation of China (51975328, 51605258), Postdoctoral Innovation Project of Shandong Province (201701011), Young Scholars Program of Shandong University (2018WLJH55) and State Key Laboratory of High Performance Complex Manufacturing, Central South University (Kfkt2017-04).

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Journal Pre-proof Fuyuan Liu: Methodology, Writing - Original Draft, Formal analysis, Data Curation Xiaoqiang Li: Conceptualization, Methodology, Writing - Review & Editing Yanle Li: Funding acquisition, Investigation, Project administration, Supervision Zijian Wang: Software, Data Curation Weidong Zhai: Data Curation, Validation Fangyi Li: Resources, Writing - Review & Editing Jianfeng Li: Methodology, Resources

Journal Pre-proof Highlights: 

The sheet forming power is proposed based on the actual contact area and sheet flow condition.

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Three sub-power models are developed and verified with good prediction accuracy.

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The effects of four process parameters on processing power, energy and efficiency are analyzed.

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The combination of process parameters that are most conducive to energy conservation is obtained.