Temperature and deformation analysis of ship hull plate by moving induction heating using double-circuit inductor

Marine Structures 65 (2019) 32–52

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Temperature and deformation analysis of ship hull plate by moving induction heating using double-circuit inductor

T

Shuiming Zhanga,b, Cungen Liua,b,∗, Xuefeng Wanga,b a b

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration(CISSE), Shanghai Jiao Tong University, Shanghai, 200240, China

ARTICLE INFO

ABSTRACT

Keywords: Ship hull plate Moving induction heating Double-circuit ODIG inductor Thermal forming behavior Transverse thermal deformations Shape parameters

This paper mainly investigated temperature and deformation behavior of ship hull plate by moving induction heating using double-circuit inductor with opposite-direction current and gap (ODIG). Mutually-coupled electromagnetic-thermal (EMT) analysis considering temperature-dependent material properties was iteratively implemented at each moving step of inductor, followed with thermal-mechanical (TM) analysis to obtain thermal deformation, and then validated by experiment. Effects of three technological parameters on thermal forming behavior (maximum temperature Tum, breadth b, depth h) and transverse thermal deformations (shrinkage δz and bending angular θz deformation) were analyzed, respectively. Besides, simplified analytical prediction for δz, θz were derived based on inherent strain and plate strip, and compared with those from TM analysis. Finally, composite quality indicator f was constructed through using Tum, b, h by PCA method and utilized to determine optimal shape parameters for ODIG based on Taguchi method, which were compared with those through using δz and θz, respectively. The results indicate that Opt-f can achieve the same f, δz, θz as those from Opt-S and Opt-B, and the best combination is C14C34T24Hy1. Therefore, f can be effectively utilized to determine optimal ODIG inductor in order to improve transverse thermal deformations (δz, θz).

1. Introduction Ship hull plates are generally consisted of complicated doubly-curved surfaces, such as saddle, concave and twisted surfaces which are typically fabricated by line heating [1] in shipbuilding. This process is mostly performed by skilled and experienced technicians through utilizing gas torch as heat source to generate non-uniform temperature distribution across thickness [2,3]. However, this inefficient manual trial-and-error method is severely dependent on experience of skilled worker resulting in low repeatability and efficiency. Meanwhile, a large amount of CO2 and noise pollution would be generated leading to significant environmental problems. Hence, alternative heat sources, such as electromagnetic induction heating (EMIH) or laser method are more predominant because these alternatives are much cleaner and more efficient. Considering small power and expensive cost for laser method [4–6], EMIH method is regarded as more preeminent heat source for large-sized ship hull plate. Considering skin effect during EMIH process and constraint of less than AC3 transformation temperature to prevent material degeneration [7], inductor is generally moved along line or triangular path, such as zigzag and fan-shaped type [8]. Recently, a few reports involving numerical analysis and experiment research have been contributed to analyzing temperature and deformation distribution for ship hull plate by induction heating. Sadeghipour et al. [9] derived two-dimensional governing equations for

∗

Corresponding author. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China. E-mail address: [email protected] (C. Liu).

https://doi.org/10.1016/j.marstruc.2019.01.001 Received 26 August 2018; Received in revised form 25 November 2018; Accepted 1 January 2019 0951-8339/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

Qi

A b

s

by B C1 C3 Cp D E Ei Em fki fk f fr g h hT Hc Hy H H Ipeak j J k l_c l_cg l_h MA m Mf MS MB N q qh qI qr Qij

Magnetic vector potential Breadth of HAZ configuration for elliptical type on top surface (mm) Breadth of HAZ at arbitrary y along thickness direction under coordinate system O-xyz as shown in Fig. 2(b) (mm) Magnetic flux density (T) One-sided breadth of Si-Fe core (mm) Upper high of Si-Fe core (mm) Specific heat capacity (J/(kg·°C)) Electric displacement (C/m2) Electric field strength (V/m) Young's modulus of mild steel at initial temperature (Pa) Young's modulus of mild steel at mean temperature Tm (Pa) kth principal component corresponding to the ith case, i = 1, 2, …, 16, k = 1,2, …,m kth principal component, k = 1,2, …,m, fk = [fk1, fk2, …, fk16]T Composite principal component Frequency of current source (Hz) Gap between inductor and plate (mm) Depth of HAZ configuration for elliptical type along thickness (mm) Convective heat transfer coefficient (W/(m2 ⋅°C)) Breadth of coil (mm) High of coil (mm) Thickness of steel plate (mm) Magnetic field strength (Wb) Peak of current source (A) Symbol of imaginary number Current density including source Js and induced current density Je (A/m2) Heat conductivity coefficient (W/(m·°C)) Length of Si-Fe core as shown in Fig. 1(a) (=80 mm) Length of whole ODIG inductor as shown in Fig. 1(a) (=120 mm) Length of heating path (=300 mm) Moving air zone above heated zone (Fig. 3) as shown in Fig. 5 Total number of quality characteristics (=3) Mean response of composite quality indicator f from Taguchi analysis Mean response of transverse shrinkage deformation δz from Taguchi analysis Mean response of bending angular deformation θz from Taguchi analysis Air around the margin of steel plate as shown in Figs. 3 and 5 Internal heat source of thermal analysis, q = qI-qrqh Convective transfer heat under air cooling Eddy current heat Radiation heat jth quality value of the ith case, i = 1, 2, …, 16, j = 1, 2, …, m

SA t T T1 T2 Th Tw Tc Tm Tum Ux Uy Uz v v V w_c w_px w_ph δhot δcool δz s z,

i z,

e z

i z,

e z

ΔT α αm β εr ε* θz s z,

ρ ρm σ σr σYm λk μ μr 33

Quality vector of the ith case, Qi = [Qi1, Qi2, …, Qim]T, i = 1, 2, …, 16 Number of principal components with their corresponding individual eigenvalues greater than one Static air zone away from heated zone (Fig. 3) as shown in Fig. 5 Variable of time (s) Variable of temperature (°C) Thickness of hollow coil (mm) Distance between two Si-Fe cores of ODIG (mm) Total Moving time of inductor, Th = 300/v (s) Sum of Th and cooling time(s) Critical temperature (=600 °C) Mean value of critical and maximum temperature at steady stage (=(Tum + Tc)/2 °C) Maximum temperature of top surface at steady stage (°C) x-direction displacement y-direction displacement z-direction displacement Moving velocity of inductor (mm/s) k kth eigenvector corresponding to the kth eigenvalue, vk = [vk1, vk2, …, vkm]T, k = 1,2, …,m Electric scalar potential (V) Breadth of ODIG inductor (mm) Breadth of air N as shown in Fig. 3 (=w_c +20 mm) Breadth of heated zone (=l_cg+20 mm) Depth of hot skin layer (mm) Depth of cool skin layer (mm) Transverse shrinkage deformation of midsurface along z coordinate direction (mm) Transverse bending shrinkage deformation δz at the starting and finishing part of heating line, respectively Temperature difference between outside surface and ambient (°C) Upper limit variable of integration in Eq. (17) Thermal expansion coefficient of mild steel at mean temperature Tm (1/°C) Lower limit variable of integration in Eq. (17) Radiation emissivity Simplified residual plastic strain Transverse Bending angular deformation of midsurface relative to z coordinate axis (rad) Transverse bending angular deformation θz at the starting, middle and finishing part of heating line, respectively Electric charge density (C/m3) Density of mild steel (kg/m3) Electric conductivity, σ1 = max(σ) (S/m) Stefan-Boltzmann constant (=5.67 × 10−8 W/m2 ⋅°C4) Yield strength of mild steel at mean temperature Tm (Pa) kth eigenvalue in descending order (k = 1, 2, …, m) Magnetic permeability, μ = μ0μr (H/m) Relative magnetic permeability, μr1 = max(μr)

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μ0 μi μm ω

s z,

e z

Free-space magnetic permeability (=4π × 10−7H/m) Poisson's ratio of mild steel at initial temperature Poisson's ratio of mild steel at mean temperature Tm Angular frequency of current source (rad/s) Edge effect on transverse bending shrinkage deformation δz at the starting and finishing part of

s z,

×

e z

heating line, respectively Edge effect on transverse bending angular deformation θz at the starting and finishing part of heating line, respectively Gradient operator Divergence operator Curl operator

induction heating treatment of steel and proposed the numerical procedure of coupled electromagnetic-thermal analysis based on finite element method. Jang et al. [10] developed an analysis method of plate forming by induction heating and analyzed the effect of various plate thickness, heat speed, input power on thermal distribution and final configuration. Shen et al. [11] established coupled mathematical model of high frequency induction heating utilizing screw coils as heat source, and then analyzed the effect of heating parameters including the distance between plate and coil, current, frequency and turns of coil on temperature distribution. Zhang et al. [12] researched the effects of technical parameters on temperature and deformation distribution for static induction heating, and obtained appropriate current frequency. They [13] also investigated edge effect under different moving ways for codirectional current-carrying inductor with no gap (CDING), including electromagnetic properties, edge temperature and shrinkage deformation distribution, and found that moving out of edge for CDING inductor can effectively eliminate edge effect. Lee et al. [14,15] investigated the effects of heating parameters including input power and moving speed on temperature and bending deformation distribution for thick plate by coupling electromagnetic-thermal analysis, and found that the dimension and microstructure of HAZ were strongly dependent on input power. Nguyen et al. [16] derived an analytic prediction of bending deformation for steel plates by triangle heating using simplified elasto-plastic model and laminated plate theory, and found that the analytic solutions were quite good compared with those from triangle heating experiment. Bae et al. [17] developed a numerical model to study triangle technique by induction heating, and analyzed the effect of heating path patterns on temperature and contraction deformation distribution. Jeong et al. [18] predicted the deformation of mild steel with initial curvature by induction heating relative to thickness and moving velocity of plate based on finite element and regression analysis. Park et al. [19] established the predictive method of angular distortion for induction heating and investigated the critical technical parameters to prevent the degradation of material properties through numerical and experiment analysis. Tango et al. [20] presented the efficient numerical TEPFE simulation technique for the plate deformation caused by induction heating, which is applicable to non-linear and repetitive heating cases, and discussed on the effect of plate edge and previous heating lines. However, few reports have been taken to analyze thermal forming behavior (Tum, b and h) and simplified analytical prediction of transverse thermal deformations (δz, θz) for ship hull plate using double-circuit ODIG inductor (in Fig. 1), and optimize combination of ODIG shape parameters in order to effectively increase δz and θz, and achieve an important step toward automatic forming process. This paper mainly investigated temperature and deformation behavior of ship hull plate by moving induction heating using double-circuit ODIG inductor as shown in Fig. 1. Mutually-coupled electromagnetic-thermal (EMT) analysis procedure considering temperature-dependent material properties was iteratively implemented at each moving step of inductor, and followed with thermalmechanical (TM) transient analysis to obtain thermal deformation. Corresponding induction heating experiment for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate was also carried out, respectively, to validate the effectiveness and determine appropriate dimensions of plate. Since longitudinal shrinkage δx and bending angular deformation θx are much less than transverse shrinkage δz and bending angular deformation θz, respectively, this paper mainly adopts transverse thermal deformations (δz, θz) as response. Then effects of three technological parameters (g, v, Ipeak) on thermal forming behavior (Tum, b, h) and transverse thermal deformations (δz, θz) were analyzed, respectively. Besides, simplified analytical method for δz, θz was derived based on inherent strain and plate strip method. Due to limited space of this paper, the accurate estimation of both edge effect and cross-heating effect would be investigated in our next research, and this paper mainly adopts thermal deformation (path C1-D1 in Fig. 3) inside the steel plate to optimize inductor shape parameters. Finally, composite quality indicator f was constructed through using Tum, b, h by means of PCA method, and then utilized to determine appropriate shape parameters for ODIG based on Taguchi method, including one-sided breadth C1 and upper high C3 of SiFe core, distance T2 between two Si-Fe cores, high Hy of coil, which were compared with those from using δz and θz, respectively. In addition, PCA method can convert multi-quality characteristics into several independent quality indicators, and part of these indicators including more than 80% information are then selected to construct a composite quality indicator f, which represents the mathematical function of the required characteristics. It can be further integrated with Taguchi method to analyze different response of each parametric variable on composite quality indicator f, and find out the optimal parameter combination through regarding composite quality indicator f as target by establishing orthogonal design of experiment (DOE). Fig. 1 presents schematic diagram of shape parameters and electromagnetic boundary conditions of half ODIG model, actual photograph of ODIG inductor and heating process for ship hull plate. 2. Analysis of ship hull plate by double-circuit ODIG inductor This section firstly analyzes thermal forming mechanism for ship hull plate fabricated by moving induction heating and brings in three thermal forming qualities, including maximum temperature Tum, breadth b and depth h of HAZ. Then governing equations of 34

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Fig. 1. Schematic diagram of shape parameters (Unit: mm) and electromagnetic boundary conditions of half ODIG model, actual photograph of ODIG inductor utilized as heat source, and heating process for ship hull plate. (a) Shape parameters and electromagnetic boundary conditions of half ODIG inductor, (b) Actual photograph of ODIG inductor and (c) Heating process, including heating direction and orientation between ODIG inductor and steel plate.

Fig. 2. Schematic diagram of dimensions (breadth b, depth h and breadth by) of HAZ configuration on top surface and cross section A-A perpendicular to heating path, respectively. (a) Breadth b of HAZ on top surface, (b) Depth h of HAZ on cross section A-A and breadth by at arbitrary y along thickness direction under coordinate system O-xyz. 35

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Fig. 3. Schematic diagram of half electromagnetic, thermal and mechanical analysis model using ODIG inductor as heat source for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate, respectively. (a) and (b) Half electromagnetic, thermal and mechanical analysis model and corresponding boundary conditions, temperature measuring points (C1, R1, S1) and deformation measuring path (C1-D1) for 500 × 400 × 16 mm3 ship hull plate, (c) and (d) Half electromagnetic, thermal and mechanical analysis model and corresponding boundary conditions, temperature measuring points (C2, R2, S2) for 600 × 500 × 16 mm3 ship hull plate.

electromagnetic analysis are derived to obtain eddy current and thus heat generation rate (HGR) distribution, which can be regarded as internal heat source for the following thermal analysis. Owing to continuously moving of inductor, mutually-coupled electromagnetic-thermal (EMT) numerical procedure is iteratively carried out to solve the governing equations above based on finite element analysis. Afterwards, thermal-mechanical (TM) analysis is established to achieve thermal deformation through using transient temperature distribution as thermal loads. Finally, in order to confirm the effectiveness of the employed numerical procedure, thermal experiment process using actual ODIG inductor (Fig. 1(b)) as heat source is fulfilled for ship hull plate (0.18% carbon) with dimensions of 500 × 400 × 16 and 600 × 500 × 16 mm3, respectively. 2.1. Thermal forming mechanism Considering mapping relationship between designed and initial flattened surface, the required deformation can be divided into membrane shrinkage and bending angular deformation. Bending angular deformation can be obtained through adjusting large temperature gradient along thickness, otherwise for membrane shrinkage deformation. When temperature distribution arrives at the critical temperature Tc of mild steel during thermal process, membrane shrinkage deformation will take place because of compressive stress generated by constraint of surrounding cool material. As a consequence, non-uniform shrinkage deformation distribution along thickness generates corresponding bending angular deformation and thus achieve required surface configuration. According to simplified thermo-elasto-plastic method [21,22], thermal deformation can be determined by maximum temperature Tum and temperature gradient along thickness. 36

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In addition, temperature gradient can be indicated by dimensions (breadth b and depth h) of HAZ configuration. Fig. 2 presents schematic diagram of dimensions (breadth b and depth h) of HAZ configuration on top surface and cross section A-A perpendicular to heating path, which can be described as elliptical and semielliptical type, respectively. When maximum temperature of top surface and dimensions of HAZ are appropriate enough, shrinkage deformation on top surface is much larger than that on bottom surface resulting in slight membrane shrinkage and larger bending angular deformation, otherwise larger membrane shrinkage and slight bending angular deformation. Besides, critical temperature Tc of ship hull plate (0.18% carbon) can be determined as 600 °C [23] to calculate the dimensions of HAZ configuration. 2.2. Numerical analysis procedure Inductor with alternating current can generate corresponding alternating electromagnetic field around and induce eddy current distribution inside adjacent mild steel [24]. The eddy current distribution can be determined through solving following differential Maxwell equations [7] as

D t

×H=J+

(1)

B t

×E=

(2)

D=

(3)

B=0

(4)

Besides, appropriate boundary conditions for solving the equations above should be adopted, commonly that is, A-V method as

B=

×A

E=

A t

(5)

V

(6)

Substitute Eqs. (5) and (6) into Eqs. (1)–(4), and take the divergence into account, then 2V

+

t

(

A) = 0

(7)

In order to simplify the differential equation above, appropriate divergence gauge, i.e., Coulomb gauge, is introduced and magnetic vector potential A can be derived by

1 µ

×

A t

×A+

Js = 0

(8)

Due to that current source is sinusoidal with time and inductor is constantly moved along heating path at given velocity, electromagnetic analysis can be approximately regarded as quasi-stationary state and therefore higher harmonics can be ignored. Then harmonic governing equation in Cartesian coordinate system can be derived from Eq. (8) and expressed as

1 µ

x

A + x y

A + y y

A y

+j

A

Js = 0

(9)

Furthermore, as for each moving step of inductor, material properties corresponding to contemporary temperature distribution would be updated and interpolated according to temperature-dependent material properties as shown in Fig. 6. Consequently, eddy current and HGR distribution can be solved under boundary conditions as shown in Fig. 1(a), Fig. 3(a) and (c), which would be subsequently utilized as internal heat source for thermal analysis. Finally, temperature distribution can be achieved through solving following heat transfer equation under boundary conditions of air cooling convection and thermal radiation as

k

T x

x

+

y

T y

+

y

T y

+q

m Cp

T =0 t

(10)

where q = qI-qr-qh, eddy current heat qI, radiation heat qr and convective transfer heat qh of air cooling can be expressed as Eq. (11). In addition, qr and qh are only applied on outside surface of steel plate except symmetric plane T as shown in Fig. 3(b) and (d).

qI =

Je2

=

( )

A 2 t

qr = r r T 4 qh = hT T

(11)

Owing to continuously-moved inductor and temperature-dependent material properties of mild steel, it is considerably formidable to obtain explicit analytical expression during EMIH process. Fig. 3 presents schematic diagram of half electromagnetic, thermal and mechanical analysis model using ODIG inductor as heat source for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate, 37

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respectively, including boundary conditions, heating direction, temperature measuring points (C1, R1, S1 and C2, R2, S2) and deformation measuring path (C1-D1). Table 1 displays coordinates of temperature measuring points for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate as shown in Fig. 3 (b) and (d), respectively, under coordinate system O-xyz. Inductor is moved from one edge to the other edge of ship hull plate along x-axis direction in Fig. 3(a) and (c). Lengths of moving path (in Fig. 3) for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate are 500 + T2 mm and 600 + T2 mm, respectively. Fig. 4 presents mutually-coupled electromagnetic-thermal and uncoupled thermal-mechanical numerical analysis procedure for ship hull plate fabricated by moving induction heating. During each iteration step, inductor is firstly moved along heating path with given displacement step (1.5 mm), and only air model MA (as shown in Fig. 5(a) and (d)) is remodeled and remeshed. Then material properties of mild steel corresponding to contemporary temperature distribution are updated and interpolated from Fig. 6 to implement electromagnetic analysis at next step for HGR distribution. Afterwards, thermal analysis using HGR as internal heat source is carried out again under conditions of air cooling and thermal radiation. This process is repeatedly implemented and circulated until inductor arrives at finial position. Hence, electromagnetic and thermal analysis are mutually coupled with each other. When inductor arrives at final position, namely t > Th, heat source would be removed and steel plate would be cooled only under conditions of air convective heat transfer and thermal radiation. After steel plate is cooled to room temperature, namely t > Tw, thermal-mechanical (TM) transient analysis is conducted to determine thermal deformation through applying transient temperature distribution at each iteration step as thermal load under boundary conditions of simply supported constraints based on von Mises criterion [3,25]. 2.3. Physical modeling analysis Since inductor is continuously moved along heating path and material properties of mild steel are strongly dependent on temperature, mutually-coupled electromagnetic-thermal (EMT) and thermal-mechanical (TM) analysis are iteratively implemented, respectively, using commercially available finite element package program ANSYS (version 16.0 64bits). Operating system and running platform for numerical analysis above are Windows 7 Enterprise 64bits, 20-cores Intel Xeon E5-2650 2.3 GHz pair with 64 GB RAM, respectively. In addition, ship hull plate is supposed to be isotropic, considerably flat and free of residual internal stress and strain. ODIG inductor is symmetrically arranged above steel plate with constant gap and moved along heating path at given velocity. Fig. 5 demonstrates half symmetric three-dimensional finite element model of electromagnetic analysis using ODIG inductor as heat source for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate, respectively, including half of two parallel-hollow rectangular coils, flowing cooling water inside coils, half of two π-type Si-Fe cores and ambient air (MA and SA). Air N (as shown in Figs. 3 and 5) around the margin of steel plate should be considered when ODIG inductor is located at initial or final location and the breadths along x axis is indicated as w_px. Meanwhile, only half of symmetric ship hull plate is considered and analyzed for thermal analysis as shown in Fig. 3(b) and (d). Because of skin effect during EMIH process, ship hull plate is partitioned into upper hot skin layer (green in Fig. 5(c) and (f)) and bottom conduction region (light blue in Fig. 5(c) and (f)) along thickness direction. The depth of hot δhot and cool δcool skin layer can be determined as [7].

500 f

hot

=

cool

= 50300

(12)

1 fµr1

(13)

1

where relative magnetic permeability μr1 and electric conductivity σ1 are the corresponding maximum values according to Fig. 6(a), namely 200 and 3.92e6, respectively. Element types employed for electromagnetic, thermal and mechanical analysis are SOLID97, SOLID90 and SOLID186, respectively. Mesh sizes of half mild steel utilized for electromagnetic, thermal and mechanical analysis are consistent just through applying transformation of element types. Since temperature field is generally less than Curie temperature (770 °C) at the beginning, a large proportion of eddy currents are still distributed among cooling skin layer and therefore a finer δcool/2-size dense mesh on top surface as shown in Fig. 5(c) and (f) is generated along thickness to ensure considerable accuracy for numerical analysis. In addition, air gap between steel plate and inductor is also densely meshed with 1 mm size to improve computational accuracy, likewise with 1.5 mm size for water, coil and Si-Fe cores. According to Lenz's law, magnetic flux density B along normal direction of symmetric plane M as shown in Fig. 3(a) and (c) is zero. Since B is indicated as the curl of magnetic vector potential A, namely × A, hence it can be equivalent to setting in-plane Table 1 Coordinates of temperature measuring points for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate as shown in Fig. 3 (b) and (d), respectively, under coordinate system O-xyz. 500 × 400 × 16 mm3 (Fig. 3 (b)) Point C1 (250,0,0)

Point R1 (280,-16,0)

600 × 500 × 16 mm3 (Fig. 3 (d)) Point S1 (250,0,30)

Point C2 (300,0,0)

38

Point R2 (330,-16,0)

Point S2 (300,0,30)

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Fig. 4. Flowchart of mutually-coupled electromagnetic, thermal and mechanical numerical analysis procedure for ship hull plate by moving induction heating under condition of air cooling.

components of A as zero, called as flux-parallel boundary conditions. Considering no magnetic leakage, flux-parallel or far field conditions are also employed on other boundaries as shown in Fig. 3(a) and (c). As for ODIG inductor as shown in Fig. 1(a), nodes of one side for coil are coupled with voltage DOF and then excited by peak Ipeak of current source, and nodes of the other side for coil are constrained with zero voltage DOF. Hence, the two coils of ODIG inductor have opposite-directional current. As for thermal and mechanical analysis, only half of symmetric mild steel model should be considered. Furthermore, radiation surface elements outside mild steel except symmetric plane T (as shown in Fig. 3(b) and (d)) are established through using SURF 152, and the emissivity is approximately 0.5 [26]. Symmetric section T is set as thermal insulation and other surfaces are constrained by air convection coefficients. As for uncoupled thermal-mechanical (TM) analysis, symmetric plane T is set as symmetric constraint and two corners of steel plate (point A and B as shown in Fig. 3(b) and (d)) are set as simply supported constraints, that is, point B is fixed by x displacement freedom Ux and y displacement freedom Uy, otherwise only y displacement freedom Uy for point A. Furthermore, temperature-dependent electromagnetic [13,27], thermal and mechanical [28–30] material properties of mild steel are presented in Fig. 6. The required calculation time for induction heating utilizing ODIG inductor as heat source is about 40 h, which is considerably time-consuming. 2.4. Experiment research In order to validate the effectiveness of the proposed procedure above and determine appropriate dimensions of plate, thermal experiment using ODIG inductor as heat source is carried out for 500 × 400 × 16 and 600 × 500 × 16 mm3ship hull plate (0.18% carbon), respectively. The actual photograph of ODIG inductor is shown in Fig. 1(b), which is moved from one edge to the other edge of ship hull plate along heating path. The gap between ODIG inductor and ship hull plate keeps constant by six universal ball bearings. Fig. 7 presents schematic diagram of experiment equipment, including induction source (EFD MINAC25/40SM), CNC controller (Galil DMC4183), laser displacement sensor (sensing head: Keyence LK-G500 and controller: LK-G3001) and temperature data acquisition (OMEGA DAQ-USB-2401), and actual heated photographs for 500 × 400 × 16 mm3 ship hull plate, respectively, under technological parameters (g = 5.0 mm, v = 2.8 mm/s, f = 12 kHz, Ipeak = 2398A). In addition, the induction source equipment is introduced from EFD induction equipment co., LTD in Norway, and it can constantly control the output of power or current effective value in ODIG inductor. Besides, the monitor can automatically display the actual power, frequency and current effective value of ODIG inductor. Coordinates of three temperature measuring points (C1, R1, S1, C2, R2, S2) are consistent with those in Table 1, among which point C1 and C2 are located on symmetric line of top surface, point 39

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Fig. 5. Half symmetric three-dimensional finite element model of electromagnetic analysis using ODIG inductor as heat source for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate, respectively, including half of two parallel-hollow rectangular coils, flowing cooling water inside coils, half of two π-type Si-Fe cores and ambient air (MA and SA). (a)–(c) 500 × 400 × 16 mm3 ship hull plate, (d)–(f) 600 × 500 × 16 mm3 ship hull plate.

R1 and R2 are located on symmetric line of bottom surface, point S1 and S2 are located on top surface with 30 mm distance away from symmetric line. Meanwhile, temperature profiles are measured through using thermocouple HH-K-24 from OMEGA ENGINEERING, INC. Finally, the ODIG inductor is controlled at constant moving velocity by DMC4183 from Galil Motion Control in USA, 40

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Fig. 6. Temperature-dependent electromagnetic [13,27], thermal and mechanical [28–30] material properties of mild steel. (a) Relative magnetic permeability and convection coefficient of air cooling, (b) Thermal conductivity and heat specific capacity, (c) Young's modulus, tangent modulus and yield's stress and (d) Poisson's ratio and thermal expansion coefficient.

which can separately control 8 servo motors with accuracy of 0.001 mm/s. Fig. 8 demonstrates the comparisons of temperature profiles for points C1, R1, S1, C2, R2, S2 (in Fig. 3), and transverse bending angular deformation θz at different x-coordinate location (in Fig. 3) for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate, respectively, obtained from between numerical analysis (FEM) and experiment (Test) under technological parameters (g = 5.0 mm, v = 2.8 mm/s, f = 12 kHz, Ipeak = 2398A). Considering that ship hull plate is gradually cooled under conditions of air cooling, the profiles just display temperature results of the first 250 s. It can be concluded that the variation tendencies obtained from numerical analysis are consistent with those from experiment. Furthermore, the relative deviations of temperature profiles for measuring points (C1, R1, S1, C2, R2, S2 in Fig. 3) and θz at different x-coordinate location (in Fig. 3) are less than 8.08% and 8.75%, respectively. Consequently, the numerical model above can be utilized to obtain maximum temperature Tum, breadth b and depth h of HAZ, and thermal deformation for ship hull plate by moving induction heating. In addition, as it can be seen in Fig. 8(d), θz for both 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate firstly increases, then remains constant and finally decreases with small magnitude along different x-coordinate location (in Fig. 3). Therefore, edge effect is mostly generated when ODIG inductor is located at initial position, which may be resulted from small temperature at the beginning. Besides, θz for 500 × 400 × 16 ship hull plate approximately remains consistent with that for 600 × 500 × 16 mm3 when x-axis location is more than 100 mm, thus the dimensions of 500 × 400 × 16 mm3 ship hull plate is large enough. Therefore, this paper mainly adopts 500 × 400 × 16 mm3 ship hull plate as example to study the temperature and deformation behavior, and then the deformation along path C1-D1 (in Fig. 3(b)) at middle x-coordinate location (250 mm) is adopted to optimize shape parameters of ODIG. 3. Analysis of temperature and deformation behavior This section firstly analyzes temperature and deformation distribution for ship hull plate by induction heating through using double-circuit ODIG inductor as heat source. Then effects of three technological parameters (g, v and Ipeak) on thermal forming behavior (Tum, b and h) and transverse thermal deformations (δz, θz) are carried out, respectively. Simplified analytical prediction for thermal deformation is established through adopting Tum, b and h based on inherent strain and plate strip assumption, which is subsequently compared with those from time-consuming TM analysis. Meanwhile, L16(45) design of orthogonal test relative to four 41

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Fig. 7. Schematic diagram of experiment equipment, including induction source (EFD MINAC25/40SM), CNC controller (Galil DMC4183), laser displacement sensor (sensing head: Keyence LK-G500 and controller: LK-G3001) and temperature data acquisition (OMEGA DAQ-USB-2401), and actual heated photographs for ship hull plate. (a) Experiment equipment, (b) Actual heated photographs for 500 × 400 × 16 and (c) 600 × 500 × 16 mm3 ship hull plate under technological parameters (g = 5.0 mm, v = 2.8 mm/s, f = 12 kHz, Ipeak = 2398A).

shape parameters of ODIG inductor is established, and corresponding thermal forming qualities (Tum, b and h) and thermal deformation (δz, θz) are obtained based on numerical model above. Finally, this section constructs composite quality indicator f through using three thermal qualities (Tum, b and h) based on PCA method. The best combination of shape parameters by Taguchi analysis through using indicator f are compared with those through using transverse shrinkage δz and bending angular deformation θz, respectively. 42

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Fig. 8. Comparisons of temperature profiles for points C1, R1, S1, C2, R2, S2 (in Fig. 3), and transverse bending angular deformation θz at different x-coordinate location (in Fig. 3) for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate, respectively, obtained from between numerical analysis (FEM) and experiment (Test), respectively, under technological parameters (g = 5.0 mm, v = 2.8 mm/s, f = 12 kHz, Ipeak = 2398A). (a) Temperature profiles of points C1 and C2, (b) points R1 and R2 and (c) points S1 and S2, and (d) θz at different x-coordinate location obtained from between numerical analysis (FEM) and experiment (Test), respectively.

3.1. Analysis of temperature and deformation distribution After obtaining the numerical analysis procedure above validated by experiment, temperature and deformation distribution for ship hull plate by moving induction heating can be carried out. Fig. 9 demonstrates temperature distribution on top surface and cross section A-A perpendicular to heating line, and dimensions (breadth b and depth h) of HAZ configuration corresponding to technological parameters (g = 5.0 mm, v = 2.5 mm/s, f = 12 kHz, Ipeak = 2512A). Fig. 9(a) describes that half temperature contour on top surface from ODIG can be approximately displayed as preheating-type “W-shape”, which can contribute to improving heating depth. The HAZ configuration above 600 °C can be regarded as elliptical shape and evaluated by breadth b of HAZ. Fig. 9(b) displays that half temperature contour on cross section A-A perpendicular to heating line can also be regarded as semielliptical shape and evaluated by depth h of HAZ. Fig. 10 presents the out-of-plane displacement Uy (Unit: m) and transverse bending angular deformation θz (rad) distribution at different x-coordinate location (in Fig. 3) on top surface under technological parameters (g = 5.0 mm, v = 2.5 mm/s, f = 12 kHz, Ipeak = 2512A). Fig. 10(a) shows that out-of-plane Uy displacement is mainly distributed in heated zone, but slightly smaller in margin of steel plate. Fig. 10(b) presents that transverse bending angular deformation θz firstly increases, then mostly remains constant and finally decreases with small magnitude at different x-coordinate location (in Fig. 3), which agrees with those obtained from experiment research in section 2.4. Fig. 11 presents transverse displacement Uz (Unit: m) and shrinkage deformation δz distribution at different x-coordinate location (in Fig. 3) under the same technological parameters as Fig. 10. Fig. 11(a) shows that transverse displacement Uz is mainly distributed in heated zone and becomes smaller among edge of steel plate. Fig. 11(b) presents that transverse shrinkage deformation δz firstly increases, then mostly remains constant and finally decreases. Hence, θz and δz have the same distribution along different x-axis position. Besides, edge effect has obvious influence on θz and δz when ODIG inductor is located at plate edge. According to the 43

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Fig. 9. Temperature distribution on top surface and cross section A-A perpendicular to heating line, and dimensions (breadth b and depth h) of HAZ configuration corresponding to technological parameters (g = 5.0 mm, v = 2.5 mm/s, f = 12 kHz, Ipeak = 2512A). (a) Temperature distribution and half breadth b/2 of HAZ on top surface, (b) Temperature distribution and depth h of HAZ on cross section A-A perpendicular to heating line.

Fig. 10. Out-of-plane displacement Uy (Unit: m) and transverse bending angular deformation θz (rad) distribution at different x-coordinate location (in Fig. 3) on top surface under technological parameters (g = 5.0 mm, v = 2.5 mm/s, f = 12 kHz, Ipeak = 2512A). (a) Out-of-plane displacement Uy distribution, (b) Transverse bending angular deformation θz (rad) distribution at different x-coordinate location.

research results by Vega et al. [1], edge effect for θz and δz can be quantified as s z

=

e z

=

s z

=

e z

=

i z

s z i z

i z

e z i z

i z

s z i z

i z

e z

(14)

i z

Hence, the effects can be calculated as 0.174, 0.024, 0.215 and 0.226, respectively. 3.2. Effect of technological parameters on thermal qualities and deformation There are many technological parameters during EMIH process for ship hull plate, including gap g between inductor and plate, moving velocity v, frequency Fr and peak Ipeak of current source. We mainly investigate the effects of gap g between inductor and plate, moving velocity v, peak Ipeak of current source on thermal forming qualities (Tum, b and h) and residual deformation (δz, θz), respectively. Table 2 presents technological parameters for different g and v under constant Tum (about 805 °C), respectively. Peak Ipeak of current source for each case is obtained through plenty of trial and error until Tum arrives at about 805 °C. Table 3 displays technological parameters to investigate the effect of different peak Ipeak of current source using ODIG inductor. Through adopting the validated numerical analysis model in section 2, thermal forming qualities (Tum, b and h) and residual 44

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Fig. 11. Transverse displacement Uz (Unit: m) and shrinkage deformation δz at different x-coordinate location (in Fig. 3) under technological parameters (g = 5.0 mm, v = 2.5 mm/s, f = 12 kHz, Ipeak = 2512A). (a) Transverse displacement Uz distribution, (b) Transverse shrinkage deformation δz distribution at different x-coordinate location. Table 2 Technological parameters for different gaps g between inductor and plate, and moving velocities v under constant Tum (about 805 °C), respectively. Case

g

v

Fr

Tum

Ipeak

Case-g1 Case-g2 Case-g3 Case-g4 Case-v1 Case-v2 Case-v3 Case-v4

3.8 4.2 4.6 5 5

2.5

12

805

1.8 2.5 3.2 4

12

805

2150 2263 2376 2512 2308 2512 2670 2840

Table 3 Technological parameters for different peak Ipeak of current source using ODIG inductor. Case

g

v

Fr

Ipeak

Case-I1 Case-I2 Case-I3 Case-I4

5

2.5

12

2263 2353 2444 2512

Fig. 12. Effect of gap g between inductor and plate on thermal forming qualities (b, h) and thermal deformation (δz, θz) under constant Tum (about 805 °C), respectively. 45

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Fig. 13. Effect of moving velocities v on thermal forming qualities (b, h) and thermal deformation (δz, θz) under constant Tum (about 805 °C), respectively.

deformation (δz, θz) corresponding to technological parameters in Tables 2 and 3 can be obtained and shown in Figs. 12–13. Fig. 12(a) and (b) demonstrate the effect of gap g on thermal forming qualities (b, h) and thermal deformation (δz, θz) under constant Tum (about 805 °C), respectively. As gap g increases, both breadth b and depth h become larger and larger, which are consistent with transverse shrinkage δz and bending angular θz deformation. These indicate that both thermal forming qualities (b, h) and thermal deformation (δz, θz) can be improved through increasing the gap g. Fig. 13(a) and (b) present effect of moving velocities v on thermal forming qualities (b, h) and thermal deformations (δz, θz) under constant Tum (about 805 °C), respectively. The breadth b keeps almost unchanged from 1.8 mm/s to 2.5 mm/s, then drops rapidly, while depth h nearly linearly decreases. Accordingly, bending angular deformation θz climbs up and then declines, but transverse shrinkage deformation δz declines rapidly. Though transverse shrinkage deformation δz decreases with moving velocity v, bending angular deformation θz can be improved from 1.8 mm/s to 2.5 mm/s. Fig. 14(a) and (b) show effect of current peak Ipeak on thermal forming qualities (Tum, b, h) and thermal deformation (δz, θz) for ship hull plate, respectively. It can be concluded that thermal forming qualities (Tum, b, h) and thermal deformation (δz, θz) can be effectively increased through adjusting current peak Ipeak. 3.3. Analysis of simplified thermal deformation Considering that the TM analysis through adopting transient temperature distribution as thermal loads to evaluate thermal deformation is time-consuming, it cannot meet high-efficiency requirement for automatic forming. Hence, thermal forming qualities (Tum, b and h) obtained from EMT analysis are utilized for quickly determining transverse shrinkage δz and bending angular θz deformation based on simplified analytical method [21,31]. These can significantly improve computational efficiency with less time to achieve an important step toward automatic forming process for ship hull plate. Breadth by of HAZ at arbitrary y-coordinate position along thickness direction under coordinate system Oxyz as shown in Fig. 2(b) can be expressed as

Fig. 14. Effect of current peak Ipeak on thermal forming qualities (Tum, b, h) and thermal deformation (δz, θz) for ship hull plate by induction heating, respectively. 46

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y h

by = b 1

2

(15)

Besides, simplified residual plastic strain ε* can be derived as

=

m Tm

1 + µi

Ym

Ei

+

1

µm

(16)

Em

where mean value Tm of critical Tc and maximum Tum temperature at steady stage inside the plastic region is employed to calculate mechanical material properties and thus simplified residual plastic strain ε*. In addition to the assumption for simplified mechanical model [21,31] that unit strip of plate perpendicular to heating line behaves like a beam, transverse shrinkage δz and bending angular θz deformation can be evaluated by adopting simplified residual plastic strain ε*among breadth b and depth h of HAZ as z z

= =

1 H

*b

12(1 H3

µi2)

y dy *yb

y dy

(17)

where the limits of integration can be determined as α = -h and β = 0 according to coordinate system O-xyz as shown in Fig. 2(b). When the depth h of HAZ is less than thickness H of plate, δz and θz can be expressed through substituting Eq. (16) into Eq. (17) as z

=

z

=

1 4 2(1

*bh H

µi2) * 3 bh 4

H3

(

H

2h

)

(18)

Finally, simplified procedure to determine thermal deformation δz, θz for ship hull plate by moving induction heating can be established, including two stages, i.e., mutually-coupled EMT analysis for Tum, b, h and subsequent simplified analytical prediction for δz, θz. According to the researches by Vega et al. [1,32] and Tango et al. [20], edge effect and cross-heating effect also should be considered to modify Eq. (18) for both δz and θz. Due to limited space of this paper, the accurate estimation of both edge effect and cross-heating effect would be investigated in our next research, and this paper mainly analyzes and evaluates thermal deformation (path C1-D1 in Fig. 3) inside the steel plate to optimize inductor shape parameters. In addition, 6 sets of arbitrary technological parameters and corresponding thermal forming qualities (Tum, b, h) obtained from EMT analysis (in section 2) are presented in Table 4. Simplified residual plastic strains ε* are also calculated through substituting Tum, b, h into Eq. (16) considering temperature-dependent material mechanical properties (as shown in Fig. 6). Meanwhile, both thermal forming qualities (Tum, b, h) and simplified residual plastic strain (ε*) are substituted into Eq. (18) to determine thermal deformation δz1 and θz1. Then they are compared with those δz2 and θz2 obtained from TM analysis method, which adopts transient temperature distribution from EMT analysis (in section 2) as thermal loads. As it can be seen in Table 5, thermal deformations δz1, θz1 from simplified procedure are almost consistent with those δz2, θz2 from TM analysis method, respectively. These indicate that simplified procedure can effectively calculate thermal deformation δz, θz with less time for ship hull plate by moving induction heating. 3.4. Analysis of inductor shape parameters In order to effectively improve thermal deformation, it is necessary to investigate appropriate shape parameters of ODIG inductor. L16(45) design of orthogonal test relative to five shape parameters of ODIG inductor (in Fig. 1) is firstly established, then corresponding thermal forming qualities (Tum, b and h) and thermal deformation (δz, θz) are obtained based on EMT (in section 2) and simplified procedure (in section 3.3). According to different requirements for each quality, b, h, δz and θz are normalized by the method “the larger is better” [33] as Ob, Oh, Oδz and Oθz, while Tum is normalized by the method “the target is better” [33] as OTum and 805 °C is regarded as the target. Since thermal deformations (δz, θz) can be evaluated by thermal forming qualities (Tum, b, h), composite quality indicator f is constructed as comprehensive assessment based on PCA method. Firstly, the obtained normalized data (OTum, Ob and Oh) is utilized Table 4 6 sets of arbitrary technological parameters and corresponding thermal forming qualities (Tum, b, h), simplified residual plastic strain (ε*), respectively. Case

g

H

Fr

Ipeak

v

Tum

b

h

ε*

1 2 3 4 5 6

4.0 4.6 5.0 4.2 4.2 4.8

18 18 18 20 20 20

12 12 12 12 12 12

2172 2534 2353 2263 2353 2399

2.8 3.6 2.5 2.5 3.2 2.5

759 767 748 800 766 786

62.28 64.92 61.44 68.78 64.22 69.72

2.65 2.83 2.87 3.04 2.95 3.81

8.71E-3 8.75E-3 8.66E-3 8.80E-3 8.75E-3 8.80E-3

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Table 5 Comparison of 6 sets of transverse shrinkage δz and bending angular θz deformation obtained from between simplified procedure and TM analysis method. Case

Simplified method θz1 2.30E-02 2.40E-02 2.24E-02 2.82E-02 2.27E-02 2.70E-02

1 2 3 4 5 6

TM analysis method θz2 2.02E-2 2.24E-2 2.12E-2 2.09E-2 1.89E-2 2.55E-2

δz1 0.2233 0.2312 0.2214 0.3048 0.2382 0.2914

δz2 0.2082 0.2328 0.2211 0.2892 0.2160 0.3048

to establish a covariance matrix, then corresponding eigenvalues and eigenvectors are calculated and indicated as λk (k = 1, 2, …, m) and vk (k = 1, 2, …, m), respectively. Since components of the kth eigenvector vk can be regarded as the weights of each quality characteristic, i.e., when the jth quality value of the ith case is indicated as Qij, the kth principal component fki of the ith case can be evaluated as a quality indicator and expressed as

fki = vk1 Qi1 + vk 2 Qi2 +

(19)

+ vkm Qim = vkT Qi

Each principal component fk interprets the variation of composite quality f at certain degree of accountability proportion (AP). When several principal components are accumulated, cumulative accountability proportion (CAP) of quality characteristics would be increased. As a consequence, this study adopts the sum of principal components with their corresponding individual eigenvalues greater than one as composite principal component f, to represent the generalized quality indicator which is defined as.

f = f1 + f2 +

(20)

+ fs

Table 6 displays L16(45) orthogonal arrays of shape parameters used for optimally designed ODIG inductor. The chosen control factors are depicted in Fig. 1, including one-sided breadth C1 and upper high C3 of Si-Fe core, distance T2 between two Si-Fe cores, high Hy of coil. In addition, breadth Hc of coil is constantly set as 10 mm and all the factors have four levels, respectively. Table 7 presents L16(45) design of Taguchi orthogonal test for shape parameters of ODIG inductor, and corresponding thermal forming qualities (Tum, b and h), composite quality indicator (f) and thermal deformation (δz, θz). Since only four factors need to be analyzed, the 5th column of L16(45) table is empty and indicated as “-”. In addition, the eigenvalues can be obtained, namely λ1 = 2.953, λ2 = 0.040, λ3 = 0.007, and the accountability proportion (AP) of first major component arrives at 98% which is greater than 80%. Hence, first major component is selected as composite principal component f, i.e. f = f1, and corresponding eigenvector v1 is equal to [0.577, 0.580, 0.575]T. Then critical shape parameters of ODIG inductor through using the indicator f based on ANOVA analysis are compared with those through using thermal deformation δz, θz, respectively, as shown in Tables 8–10. It can be concluded that only one-sided C1 and T2 significantly affect composite quality indicator f, transverse shrinkage δz and bending angular deformation θz under confidence level of 0.01. Finally, the best combination of shape parameters for ODIG inductor can also be achieved based on Taguchi analysis through regarding composite quality indicator f, transverse shrinkage δz and bending angular deformation θz as target response, respectively. Fig. 15 presents mean responses Ms, Ms, Mb of composite quality indicator f, transverse shrinkage deformation δz, bending angular deformation θz relative to each shape parameter, respectively. As it can be seen, mean response Mf for f (in Fig. 15(a)) is absolutely consistent with both MS for δz (in Fig. 15 (b)) and MB for θz (in Fig. 15 (c)). The best combination for f, δz and θz is C14C34T24 Hy1 (Opt-f, Opt-S), meaning that C1 = 7.0 mm, C3 = 7.0 mm, T2 = 13.0 mm and Hy = 8.0 mm. Table 11 summarizes the initial (Opt-I) and the obtained combination (Opt-f, Opt-S and Opt-B) of shape parameters for ODIG inductor, and corresponding thermal forming qualities (Tum, b and h), composite quality indicators (f) and thermal deformations (δz, θz). It can be found that the obtained inductor Opt-f can achieve the same quality indicators (f) and thermal deformations (δz, θz) as those from Opt-S and Opt-B, which are much larger than those from Opt-I. Hence, optimal shape parameters of ODIG inductor can be perfectly achieved through using composite quality indicator f, in which only EMT analysis needs to be carried out for Tum, b and h, and thus effectively reduce computational time. The procedure can also be utilized to search the optimal combination of shape parameters for other inductor structures, in order to improve thermal deformations (δz, θz). Table 6 L16(45) orthogonal arrays of shape parameters used for optimally designed ODIG inductor. Factor

Unit

Levels

Values

C1 C3 T2 Hy

mm mm mm mm

4 4 4 4

5.5 6.0 6.5 7.0 5.5 6.0 6.5 7.0 7 9 11 13 8 9 10 11

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Table 7 L16(45) design of Taguchi orthogonal test for shape parameters of ODIG inductor and corresponding thermal forming qualities (Tum, b and h), composite quality indicator (f) and thermal deformations (δz, θz). Case

C1

C3

T2

Hy

–

Tum

b

h

f

δz

θz

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

5.5 5.5 5.5 5.5 6.0 6.0 6.0 6.0 6.5 6.5 6.5 6.5 7.0 7.0 7.0 7.0

5.5 6.0 6.5 7.0 5.5 6.0 6.5 7.0 5.5 6.0 6.5 7.0 5.5 6.0 6.5 7.0

7 9 11 13 9 7 13 11 11 13 7 9 13 11 9 7

8 9 10 11 10 11 8 9 11 10 9 8 9 8 11 10

1 2 3 4 4 3 2 1 2 1 4 3 3 4 1 2

698 721 737 745 745 720 784 771 782 794 763 787 809 808 793 785

49.96 55.50 59.26 61.88 61.58 56.40 70.19 66.58 70.12 71.04 64.84 70.72 77.38 76.58 73.16 71.10

1.78 2.22 2.59 2.83 2.82 2.21 4.03 3.64 3.87 3.34 3.43 4.01 4.64 4.54 4.22 3.96

0.5950 0.8167 0.9726 1.0695 1.0643 0.8164 1.4911 1.3458 1.4592 1.4659 1.2612 1.5064 1.7097 1.6981 1.5833 1.4912

0.192 0.211 0.242 0.270 0.265 0.217 0.355 0.318 0.358 0.391 0.301 0.361 0.443 0.434 0.394 0.362

2.24E-2 2.38E-2 2.50E-2 2.58E-2 2.58E-2 2.42E-2 2.81E-2 2.73E-2 2.82E-2 2.88E-2 2.70E-2 2.83E-2 2.94E-2 2.93E-2 2.88E-2 2.84E-2

“-” represents that the control factor is empty Table 8 Results of ANOVA analysis carried out for shape parameters of ODIG inductor through using composite quality indicator f. SV

DOF

SS

MS

F

P

C1 C3 T2 Hy Error Total

3 3 3 3 3 15

1.27938 0.07652 0.35959 0.01935 0.01137 1.74621

0.426461 0.025507 0.119863 0.006450 0.003788

112.57 6.73 31.64 1.70

1.399E-3 0.076 9.020E-3 0.336

SV (Source of variation); DOF (degrees of freedom); SS (Sum of squares); MS (Mean square); F (F-value); P (Probability). Table 9 Results of ANOVA analysis carried out for shape parameters of ODIG inductor through using transverse shrinkage deformation δz. SV

DOF

SS

MS

F

P

C1 C3 T2 Hy Error Total

3 3 3 3 3 15

0.369427 0.002955 0.104600 0.007612 0.000854 0.485447

0.123142 0.000985 0.034867 0.002537 0.000285

432.80 3.46 122.54 8.92

1.878E-4 0.167 1.233E-3 0.053

Table 10 Results of ANOVA analysis carried out for shape parameters of ODIG inductor through using bending angular deformation θz. SV

DOF

SS

MS

F

P

C1 C3 T2 Hy Error Total

3 3 3 3 3 15

0.060164 0.003401 0.016674 0.000185 0.000445 0.080868

0.020055 0.001134 0.005558 0.000062 0.000148

135.30 7.65 37.50 0.42

1.064E-3 0.064 7.052E-3 0.755

4. Conclusions This paper mainly investigated temperature and deformation behavior of ship hull plate by moving induction heating using ODIG inductor as heat source. Mutually-coupled EMT analysis considering temperature-dependent material properties was iteratively implemented at each moving step of inductor, and followed with TM analysis to obtain thermal deformation, and validated by experiment research. Then effects of three technological parameters (g, v, Ipeak) on thermal forming qualities (Tum, b, h) and transverse thermal deformations (δz, θz) were analyzed, respectively. Besides, simplified analytical prediction for δz, θz were derived 49

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Fig. 15. Mean responses (Mf, MS, MB) of composite quality indicator f, transverse shrinkage δz and bending angular deformation θz relative to each shape parameter for L16(45) orthogonal test based on Taguchi analysis, respectively.

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Table 11 Comparisons among the initial (Opt-I) and best combinations (Opt-f, Opt-S and Opt-B) of shape parameters for ODIG inductor, and corresponding thermal forming qualities (Tum, b and h), composite quality indicator (f) and thermal deformation (δz, θz). ODIG

C1

C3

T2

Hy

Tum

b

h

f

δz

θz

Opt-I Opt-f Opt-S Opt-B

5.5 7 7 7

6.5 7 7 7

13 13 13 13

10 8 8 8

750 808 808 808

59.9 78.5 78.5 78.5

2.93 4.68 4.68 4.68

1.0931 1.7292 1.7292 1.7292

0.275 0.462 0.462 0.462

2.60E-2 2.96E-2 2.96E-2 2.96E-2

based on plate strip method. Finally, composite quality indicator f was constructed through using Tum, b, h based on PCA method and then utilized to optimize shape parameters of ODIG inductor based on Taguchi method, which were subsequently compared with those from using δz and θz, respectively. Conclusions can be summarized as follows: ● Temperature profiles (points C1, R1, S1, C2, R2, S2) and θz at different x-coordinate location for 500 × 400 × 16 and 600 × 500 × 16 mm3 ship hull plate obtained from the proposed procedure are consistent with those from experiment, and relative deviations are within 8.08% and 8.75%, respectively. Temperature distribution can be regarded as preheating-type “Wshape”. Since θz for 500 × 400 × 16 ship hull plate almost remains consistent with that for 600 × 500 × 16 mm3 when x-axis location is more than 100 mm, the dimensions of 500 × 400 × 16 mm3 ship hull plate is large enough. ● g, v and Ipeak have significant effects on Tum, b, h, δz and θz, respectively. Transverse thermal deformations (δz1, θz1) from simplified procedure are almost consistent with those (δz2, θz2) from TM analysis method. Due to limited space of this paper, accurate estimation of both edge effect and cross-heating effect would be investigated in our next research, and only thermal deformation (path C1-D1 in Fig. 3) inside the steel plate is utilized to optimize inductor shape parameters. ● Accountability proportion (AP) of first principal component arrives at 0.978 which is greater than 0.8, and the other two eigenvalues are both less than 1. Therefore, the first principal component f1 can be regarded as composite quality indicator f. Only the probabilities of C1 and T2 from ANOVA analysis are no more than 0.01, thus C1 and T2 are the critical factors for ODIG inductor. ● The best combination for f (Opt-f), δz (Opt-S) and θz (Opt-B) is C14C34T24 Hy1. Opt-f can achieve the same quality indicators (f) and thermal deformations (δz, θz) as those from Opt-S and Opt-B, which are much larger than those from Opt-I. Hence, composite quality indicator f can perfectly evaluate transverse thermal deformations (δz, θz) to achieve optimal ODIG inductor, which doesn't have to carry out time-consuming TM analysis. Acknowledgements This work has been financially supported by the National Key Basic Research and Development Program (Project No. 2013CB036103); and Research Project of State Key Laboratory of Ocean Engineering, Development of Intelligent Machining Robot for Large Curvature Ship Plate (Project No.GKZD10010). References [1] Vega A, Osawa N, Rashed S, et al. 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Temperature distribution and bending behaviour of thick metal plate by high frequency induction heating. Mater Res Innovat 2011;15(sup1):s283–7. [15] Lee KS, Eom DH, Lee JH. Deformation behavior of SS400 Thick plate by high-frequency-induction-heating-based line heating. Met Mater Int 2013;19(2):315–28. [16] Nguyen TT, Yang YS, Bae KY. Analysis of bending deformation in triangle heating of steel plates with induction heating process using laminated plate theory. Mech Base Des Struct Mach 2009;37(2):228–46. [17] Bae KY, Yang YS, Hyun CM. Analysis of triangle heating technique using high frequency induction heating in forming process of steel plate. Int J Precis Eng Manuf 2012;13(4):539–45. [18] Jeong CM, Yang YS, Bae KY, et al. Prediction of deformation of steel plate with forced displacement and initial curvature in a forming process with high frequency induction heating. Int J Precis Eng Manuf 2013;14(5):785–90.

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