1 Introduction

Induction heating is a complex process that consists of several physical phenomena including thermal, electrical, and magnetic fields. In this process, the heat is generated on the surface of a magnetic object when it is exposed to an altering magnetic field. In fact, eddy currents derived from altering magnetic fields, flow over the object, and inherent electrical resistant of the material leads to heat generation in the object [1]. This method has proven its reliability for being used in mass productions. Induction heating for industrial purposes offers a fast generating of high-intensity heat at a well-defined location in parts with repeatable quality, low labor cost for operating induction machine, being energy-efficient and ecofriendly. Furthermore, it makes industries capable to treat metallic parts with complex-geometric shapes such as gears and splines. All these advantages lead to delivering fast products with desirable and constant quality and lower cost. Induction heating can be applied to annealing, bonding, brazing, carbide tapping, casting, curing and coating, forging, melting, plastic reflow, heat staking preheat and postheat, shrink fitting, and soldering [2]. Besides, this method is widely used for case hardening, which is also known as surface hardening. The case hardening of a spur gear using the induction case hardening process is illustrated in Figure 1 wherein the yellow surface is induction-heated. The machine is equipped with a cooling system, that cools the gear rapidly after heating. This fast cooling increases the generation of martensite phase, in the heated area resulting in a more hardened gear surface compare to the gear core.

Fig. 1
figure 1

Induction hardening of a spur gear [3]

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Generally, spur gears are subjected to altering forces that are the reason for gears’ fatigue failure. Increasing wear-resistant in contact areas resulting in higher contact fatigue strength. One simple solution to this issue is quenching heat treatment, in which the whole part is undergone hardening heat treatment. This increases the martensite phase uniformly in a part, both on the surface and in the core, resulting in a brittle gear, which is not a desirable mechanical property for gears. In this regard, case hardening heat treatment would be a credible choice by providing a fine martensite phase only at the surface layers of the gear. Applying case hardening results in a hard gear surface without changing the hardness and metallurgical microstructure of the gears’ core [2, 4]. In general, the result of case hardening highly depends on the components of steel alloys. Overall, steel alloys are categorized by the percentage of constituent alloys into low, medium, and high-alloy steels [5]. In this study, AISI 4340 steel is used, which is low alloy steel presenting the utmost possible toughness and strength in a heat-treated condition, as well as good fatigue resistance. In this study, the purpose of case hardening of AISI 4340 steel gear is to change, to the greatest degree, the perlite and bainite phase to martensite to enhance the surface hardness of gears and to provide fine martensite microstructure. To prevent producing any pearlite or bainite phases during cooling process, the austenite layer should be crystallized uniformly, during heating process [6]. A desirable austenite layer formation could be fulfilled by choosing proper process parameters in induction hardening in which the remaining martensite layer is guaranteed by using a suitable quenching method such as a water shower. Conventional heat treatment procedures for AISI 4340 alloy steel parts is performed by heating the part within a range of 790 to 915 °C (850 °C in our study) to achieve a uniform austenite structure in its surface layer, then a rapid quenching to create maximum possible martensite layer [7,8,9,10].

Induction hardening of gears engages with several challenges. The major challenge is in induction heating wherein the edge effect is brought about due to the variable density of magnetic flux at different areas of a part [2, 11]. This variation in flux density causes diverse temperature map through the heated area. Generally, the hardness profile obtained from hardening methods is not uniform. A typical quenching heat treatment creates a nonuniform martensite layer on the gear surface resulting from a nonuniform metallographic structure through the thickness of the part. In the case of induction hardening of external spur gears with the single-shot method and high-frequency current, the nonuniformity in temperature could be seen in two directions: radial and axial direction. The nonuniformity in axial direction causes a difference of temperature between the root and tip of the gear tooth that is due to the proximity of tooth tips to the induction coil compared to tooth roots. In axial direction, the issue referees to the fact that the generated heat between the middle plane and edge plane of a gear tooth, as shown in Fig. 2a, is not uniform. This is termed “edge effect”, which happens when the current is induced in thin bodies like slabs, stripes, foils, and thin gears in which the thickness of a part is considerably less than its width and length. When eddy currents bend to reverse direction in corners of close areas like gear edges near the end of corners, distortion of current takes place since current tends to break the corners. Overheating or underheating of these specific areas can create an uneven heat profile resulting in a nonuniform martensite layer on the gear [5, 13, 14]. The edge effect of a heat-treated gear using induction heating method is illustrated in Fig. 2b and 2c wherein a different hardness profile between the edge plane of gear tooth (Fig. 2b) and the middle plane of gear tooth (Fig. 2c) can be observed. It is worth mentioning that the most case depth difference between two planes (middle and edge plane) happens at the teeth root of gears.

Fig. 2
figure 2

a Schematic representation of the edge plane and middle plane, and the case depths in b edge plane [12] and c middle plane [12] of a gear

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To the best of the authors’ knowledge, there is no technology to eliminate this undesirable effect. However, efforts are done on hindering it as much as possible by utilizing flux concentrators. Process arrangements such as axial and radial gaps (Fig. 4) are of importance in minimizing the edge effect. Therefore, minimizing the edge effect demands optimizing the induction heating machine parameters including machine power, frequency, heating time, and the axial and radial gaps. Over the last few decades researchers attempt to optimize induction heating process parameters with methods such as designing different-shaped coils, using variable power supplies, quenching systems for various part geometries to achieve desirable hardened layer microstructure and case depth. Since the early years of thriving industrial experiences up to now, induction heating technology has seen many improvements such as creating solid-state power supplies instead of motor generator and development of high current, which in their turn have caused more efficient processes with lower costs [2]. Studies are also carried out on the other objectives in the induction hardening field such as finding a relationship between fatigue tooth breakage start point and machine parameter, and case depth using power circulating gear testing machine for steel spur gear [15]. Zhichao et al. [16] investigate the effect of preheating of parts on the residual stress and cracking during quenching of large parts of 4340 steel. Barglik [17] presented the spatial temperature distribution through a dual frequency induction heated gear and validate the model by experimental studies. Kristoffersen et al. [18] presented research that studies the relationship between residual stress and hardening depth for two microstructures of cylindrical parts by using an X-ray diffraction method. Cajner et al. [19] studied the effects of induction hardening machine parameters on cylindrical workpieces using computer simulation for an electromagnetic process, thermodynamic, microstructural transformations, and hardness distribution in a part. Residual stresses of surface hardened steel parts using induction heating method are also investigated both experimentally, using X-ray diffraction technique, and numerically, using multiphysics finite element modeling [20]. Candeo et al. [21] conducted computer simulations with electromagnetic, thermal, and metallurgical finite element analyses for induction hardening of spur gears. Munikamal et al. [22] exploited ABAQUSTM simulation software to determine the case depth and hardness in surface hardening of automotive steel components such as crankshaft and spur gears. Montalvo et al. [23] used computer simulation to demonstrate the effect of quenching rate on transformation-induced plasticity (TRIP) effect in a high-frequency induction heating process wherein results showed that the effect is not negligible. Barglik et al. [24] established temperature profiles and the final hardness of a gear wheel using 3D numerical analysis with temperature-dependent parameters of the hardening process. Other research works have also been carried out using 3D simulation to analyze and predict AISI 1045 steel behavior during spot induction hardening (SIH) [25], and to estimate residual stresses in small spur gears hardened by induction heating [26]. Moreover, 3D simulations of multifrequency induction hardening were accomplished for a mathematical model including Maxwell’s equations and nonlinear heating equation to eventually link thermal and electromagnetic equations [27]. Computer simulations are also used to determine temperature distribution in a workpiece in multifrequency-induction-hardening with a single coil for parts with complex geometries [28], and to predict material properties for spot continual induction hardening [29]. Recent researches [30,31,32] are fulfilled to develop numerical studies on dual-frequency hardening process of AISI 300 M gear. They introduce a mathematical model of the process with the help of time-temperature- austenitization (TTA) and continuous-cooling-temperature (CCT) diagrams [33]. Marco Baldan et al, investigated the induction heating process from another point of view, wherein multifidelity optimization approach is implemented to reduce process time [34]. Li et al. [16] also tried to optimize the induction heating temperature of 55CrMo Steel ball screws in which they used the martensite fraction value as main criterion of their study. The outcome of the study shows that the more temperature, the more hardness and the more martensite volume fraction generation [35].

To investigate the effects of heat treatment process parameters on pipes residual stress in medium-frequency induction, 3D analysis of medium-frequency induction heating of steel pipes are conducted [36]. More recently, Xiaobin et al. [37] carried out numerical and experimental studies on local induction heating of large-diameter gear rolling. It is worth noting that Tong et al. [38] applied a couple of simulations and numerical models together to investigate residual stress distributions of gear tooth in an asynchronous dual-frequency (ADF) induction hardening process for spur gears. Our research group implemented new techniques to accomplish enhancement in the surface hardening of gears. We used a 3D model simulation of spur gears to the determinate extensive effect of machine parameters and coil geometry on the final temperature distribution on specimens [39]. The most effective machine parameters on the case depth of spur gears [40] and the effects of machine parameters on the case depth of flat cylinders by using computational simulation [5] were studied. More recently, we studied the effect of flux concentrators on the case depth of gears using experiments, analytics, and simulation software to achieve a more efficient method. In this regard, we investigated using COMSOLTM simulation for a 3D finite element model on spur gears [5]. We further evaluated the effects of flux concentrators on the edge effect of gear tooth [12]. Researches studying the effect of flux concentrators on the final temperature distribution and case depth in the induction heating of spur gears using finite element analysis, analysis of variance (ANOVA), and artificial neural network methods, are also conducted to predict models leading to the edge effect reduction [41, 42].

Although preceding researches have investigated the case depth of simple and symmetrical shapes, edge effect control in complex shapes such as gears is still a challenge. To the best of the authors’ knowledge, no study analyzes and prioritizes the influence of induction machine parameters on the case depth of spur gears in both experimental approach and statistical analyses. In this research, an experimental method is adopted, and results are explored using ANOVA to analyze the effect of induction surface heating parameters on the case depth. This statistic approach is planned, to perform most precisely yet with the optimum number of tests. This proposed statistical approach is developed in progressive steps. The first step is to prepare an experimental plan according to Taguchi planning method [43]. The second step is involved in defining the induction hardening ranges for spur gears. The third step is consisted of acquiring the experimental results and analyzing the data using ANOVA [44]. Based on ANOVA analyses and linear regressions, predictive models are developed to assess the case depth in the tip and root of gear teeth. This model is executed to achieve an optimized condition in the induction heating process for the case depth of spur gears. The effect of flux concentrators on edge effect and the consequences of using them in induction heating of thin spur gears are also investigated in this study. The feasibility and effectiveness of the proposed approach led to an accurate and reliable statistical model for predicting the case depth of spur gears. This article is formed as follows: Section 2 presents the methodology of our experimental tests based on induction surface heating and its related parameters. Results of experimentation, statistical and sensitivity analyses, and edge effect discussions are then reported in Section 3. In the end, Section 4 presents conclusions and ideas for future works in this field.

2 Experiment procedure

2.1 Materials and induction heating process

The induction heating experiments in this study have been performed on spur gears made of AISI 4340 steel, that are used widely in power transmission gears due to high strength in heat-treated condition and excellent fatigue strength. The chemical composition of AISI 4340 steel and its mechanical characteristics are presented in Tables 1 and 2 respectively.

Table 1 AISI 4340 steel chemical composition in wt% [45]
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Table 2 Mechanical properties of AISI 4340 steel Q and T [46]
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Induction hardening tests were carried out by the induction machine illustrated in Fig. 3. The machine exploits two generators to produce a varying range of frequencies from medium to high frequency. The frequency rate is controlled by generator power in a way that the maximum frequency will be achieved by the machine’s maximum power, which in its turn is obtained by combining the maximum power of two generators. In order to reach the maximum frequency, first, the audio-frequency of 10 kHz is obtained at the maximum generator power of 550 KW using solid-state converters. Then, 200 kHz frequency is generated by a radiofrequency (RF) transistor generator at the maximum power of the second generator in 450 kW. In this study, the radiofrequency (RF) transistor generator operates at 200 kHz. The real-time monitoring (RTM) module is also applied to ensure the quality of the process in a way that the parameters are checked whether the actual parameters are the same as the operator’s intended parameters.

Fig. 3
figure 3

Induction machine and operation system [47]

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The schematic of a spur gear and flux concentrators is illustrated in Fig. 4. Two flux concentrators are gears exactly similar to the master spur gear. They have positioned above and under the spur gear employing an available stand kit and using AISI 1010 steel shims to ensure axial gaps. The thickness and the outer diameter of these shims are 0.2 ± 0.05 mm and 31.75 mm respectively. The master spur gear is surrounded by a coil with 140 mm of outer diameter, 110 mm of inner diameter, a useful section of 7 × 7 mm, and 2 mm thickness. Finally, the machine is equipped with a cooling system comprising a jet of coolant solution containing 92% water and 8% polymers. The quenching heat treatment is performed by a cooling system which is situated under the gears as shown in Fig. 3.

Fig. 4
figure 4

Schematic presentation of the model’s geometry

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All used gears in this research are made of AISI 4340 steel with identical shape and geometry with an outer diameter of 104.3 mm, and a thickness of 6.5 mm. To make sure that gears are containing homogenous mechanical and microstructural properties, the gears were undergone quenching and tempering heat treatments. Using microhardness tests the uniform hardness of gears reaches approximately 40 HRC in this step. It should be noted that due to the geometrical complexity of gears, the roots and tips of gear teeth are exposed to different magnetic flux densities. To prevent overheating, the temperature should not exceed the austenite transformation temperature. As already mentioned, induction heating of gears is a function of the process parameters. To avoid redundancy and to find an optimum number of parameter variations for experimental tests, a design of experiment method is implemented. The efficiency and simplicity of factorial design and more specifically Taguchi planning method, make it the most commonly used method allowing choosing factor levels and simultaneous mode of factor variations, studying the effect of each factor on the process [43, 48]. To predict experimentally the case depth in tip and root of spur gears, Taguchi method is used to minimize the number of our tests while providing expressive results for statistic analyses.

2.2 Design of experiments based on Taguchi method

The generated heat is adjustable by machine input variables consisting of machine power, heating time, and frequency. However, modifying frequency in induction heating is not as straightforward as machine power and heating time. The concentrator gap also plays an important role in controlling the induction heating of gears. The axial and radial gaps, the most important geometrical factors of the process, are determined as a gap between the gear and coil as well as the gear and concentrator respectively (Fig. 4). In this research, the axial gap is considered as a study term while the radial gap is preserved as a constant at 2.3 mm. In this regard, researches on the variation of radial gap and its effect could be the subject of investigation in further studies. Regarding Taguchi method, three (3) levels for each parameter are considered to cover all practical parameter ranges including low, medium, and high levels which is presented in Table 3. The controlling parameters are power (P), heating time (T), and the gap between a concentrator and the master gear (G).

Table 3 Experimental factors and range of variables for each level applied to Taguchi method
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As already mentioned, to develop a reliable analytical model through analysis of induction heat treatment experiments, it is essential to carry out a properly designed test plan to perform a minimum efficient number of experimentations using Taguchi experiment planning. The suitable range of factors along with the Taguchi experimental design makes the process more robust, efficient, and with a more consistent performance in a minimal number of trials [49, 50]. In the case of induction hardening processes, it is important to search for the variation of experimental parameters allowing minimal and maximal case depth transformation. To this end, the maximal and minimal parameter range tuning is necessary before establishing parameter levels. This is defined during our preliminary tests as mentioned before in this section. Based on Taguchi planning method, the design of experiments that addresses this experiment is an L9 orthogonal matrix corresponding to nine (9) tests. Consequently, each test consists of a unique combination of parameter levels which are shown in Table 4.

Table 4 Design of experiences, L9 orthogonal matrix, and experimental results
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2.3 Case depth measurement

In order to measure the case depth of gear after the case hardening process, microindentation tests using Clemex microhardness [51] testing machine is performed on the edge and middle planes of gears (Fig. 2a). Microindentation tests are repeated on each of nine induction heated gear to evaluate the effect of process parameters on case hardening. These results are shown in Table 5 which presents the case depth measurement at gears tooth root in the middle plane (dRM) and the edge plane (dRE) as well as at gears tooth tip in the middle plane (dTM) and the edge plane (dTE).

Table 5 Microhardness test results of the tooth root (R) and tooth tip (T); hardness has been measured on the middle plane (M) and the edge plane (E)
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It can be observed in Table 5 that increasing P and T may result in reducing dR; however, no significant effect could be seen for dT. These results imply that the maximum and minimum case depth occurs in tests 3 and 1 respectively. The hardness profile of the gear on tip and root area resulting from tests 1 and 3 are compared in Figs. 5 and 6. Three regions could be noticed in theses graphs: the first region presents a hardened surface layer due to martensitic microstructure formation, the second region displays a hardness drop to approximately the core material hardness, and the third region preserves the initial hardness of gear as it did not receive the thermal flow [52]. Figure 5a demonstrates the maximum and minimum obtained case depth at middle-root (dRM) resulted respectively from test 3 and test 1. Figure 5b shows the maximum and minimum case depth at edge-root (dRE). In Fig. 6a, the maximum and the minimum attained case depths at middle-tip (dTM) of test 1 and test 3 are compared together. Moreover, the comparison between the maximum and minimum case depth at edge-tip (dTE) is presented in Fig. 6b.

Fig. 5
figure 5

Microhardness of teeth root for test 1 and test 3 (a) in middle plane dRM (b) in edge plane dRE

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Fig. 6
figure 6

Microhardness of teeth tip for test 1 and test 3 (a) in middle plane dTM (b) in edge plane dTE

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Three main conclusions can be drawn from Table 5. First, the maximum case depth for both root and tip can be achieved at the highest machine power and the longest heating time that are 170.0 kW and 0.55 s respectively. In contrast, the minimum case depth can be carried out at the lowest level of parameters (143.0 kW and 0.45 s). This is due to the fact that the more heating time, lets the heat to convey through more surface, resulting in a more heated area. Second, in each gear, generally, the case depths in the middle plane are deeper than the case depth in the edge plane. This is an affirmation of generating more current density at the middle of teeth compared to the edges due to the induction edge effect. Third, a proper variation of collected case depths in terms of having fine coverage between the maximum and minimum depths in roots and tips, remarks on the credibility of well-designed tests with Taguchi method.

3 Effect of parameters on hardness profile

3.1 Statistical analysis and parameters contribution

To investigate the effect and influence of each parameter on the case depth of spur gears, analyses of variance (ANOVA) are applied [53, 54]. The exactitude of the regression model is evaluated by an F test based on ANOVA. General systematic details are presented in Tables 6, 7, 8, and 9. Terms that present the results of the analyses are degrees of freedom (DoF), sum of squares, contribution percentage, F value, and P value [53, 55]. To determine whether the effect of process parameters on the case depth is statistically significant, ANOVA calculates p value based on F test. Comparing p value with the designated significance level (α) determines whether the null hypothesis can be rejected or not. The null hypothesis, in this case, states that the effect of studied parameters is not significant on the case depth. We reject the null hypothesis when the calculated p value is less than the designated significance level. In this study, we use a significance level (α) of 0.05 which indicates that there is a 5% risk of concluding that a parameter has a significant effect on the process when there is no actual effect.

Table 6 Results of ANOVA for micro-hardness tests at root and on middle plane (dRM.) of gear
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Table 7 Results of ANOVA for micro-hardness tests at root and on edge plane (dRE) of gear
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Table 8 Results of ANOVA for microhardness tests at tip and on middle plane (dTM) of gear
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Table 9 Results of ANOVA for microhardness tests at tip and on edge plane (dTE) of gear
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In this study, linear regression is developed based on ANOVA results wherein the case depths are estimated as a dependent variable (dRE, dRM, dTE, and dTM), and power (P), heating time (T), and concentrator’s gap (G) as independent variables. Preliminary effect analyses for each of the dependent variables (dRE, dRM, dTE, dTM), using interaction diagrams of factors in three levels, show that there are no significant interactions between independent variables. In this regard, adding factor interactions into ANOVA model can use the DoF up, and make the model complicated and incalculable [56]. During preliminary ANOVA analyses, first-order interaction of factors (T×P, T×G, and P×G) are added to the ANOVA model along with the factors (P, T, and G) to determine the significance of their effect on dRM, dRE, dTE, and dTM. These results showed that the calculated p value for interactions was much higher than the significance level (0.05), which infers that the interactions are not significant in these cases. These large p values raise a red flag for an overfitted model which can consequently lead to erroneous predictive models [57]. For this reason and to provide a reliable goodness-of-fit regression model, ANOVA analyses for dRM, dRE, dTE, and dTM are performed without the interaction of independent factors.

Table 6 shows ANOVA analyses for dRM in which the p value for P, G, and T are respectively 0.004, 0.008, and 0.017. These p values are less than 0.05, hence, the null hypothesis of nonsignificant factors for these terms is rejected. It can be observed that machine power (P) is the most effective parameter, then heating time (T), and finally concentrator’s gap (G) with respect to their incremental quantity of p values. Thinking back to p values for dRM and considering the contribution percentage variables, it can be concluded that P is the most effective term in dRM. In the same way, for dRE in Table 7, P and T respectively are the most effective parameters. Considering contribution percentage of 53.88% and p value of 0.001for dRE declares that P is the most effective term in dRE.

Table 8 for dTM remarks p value of 0.000 for P and 0.001 for T. p value of 0.000 for P implies that the null hypothesis is rejected with high confidence. Considering a contribution of 52.32%, the effect of P in dTM is very significant. p value of 0.001 and the contribution of 36.33% for T shows also the significant effect of this parameter. Results in Table 9 expresses that in dTE, the p value of T is 0.000, and of P is 0.001. This indicates that T is the primary effective parameter and P is the secondary effective term in dTE. It should be noted that the effect of G was not significant enough to be considered as a contributing factor, and this factor is eliminated during iterations of ANOVA.

3.2 The main effect of process parameters

For easier correlation determination between dependent and independent variables, statistical methods are used to specify the effect trend of each parameter on the response. Using the sum of squares calculated for each parameter in ANOVA tables, the contribution of each factor in the total sum of squares indicates the effect of parameter values on the response variation. To this end, the results of ANOVA analysis are used to determine the contribution ratio of each parameter (P, T, and G) in the case depths for root (dRE, and dRM) and tip (dTE, and dTM) of the gear. As shown in Tables 6, 7, 8, and 9, the contribution percentage of each process parameter (P, T, and G) confirms in most cases, that the highest contribution percentages belong to the parameters P and T, while G has a minimum influence on the case depth. The contribution percentages of P are 40.83%, 53.88%, 52.32%, and 41.91%, and of T are 30.00%, 32.59%, 36.33%, and 51.51% for dRM, dRE, dTM, and dTE in turn. These contribution percentages concerning G are 20.83%, 8.15%, and 8.20%, for dRM, dRE, and dTM respectively.

In order to have a better visual perspective of the impact of each parameter on responses, the main effect plots are drawn in MATLABTM, in which the statistical mean value of responses in each level of parameters presents the trend of each parameter on the response. In this regard, the steeper is the plot, the more influencing is the studied parameter. The main plots present separately the effectiveness of each parameter at varying levels. Through these plots, each level of P, T, and G are linked to their response mean value at the parameter level. Hence, to examine differences between level means of independent variables (P, T, G) on each of four responses (dRE, dRM, dTE, and dTM), main effect plots are presented in Fig. 7a and b.

Fig. 7
figure 7

Main effect plots of case depth on middle and edge planes at a root b tip

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Figure 7a is dedicated to show the case depths of tooth root where dRM and dRE are respectively distinct by red and blue lines. Generally, the steepness of lines from a level to the next level shows the effect of each parameter on the case depth response; the steeper the line, the more effective the parameter on case depths. It could be observed in Fig. 7a, for dRM with red lines, that P and T are the most effective parameters over the three levels due to the biggest range of the case depth from the first to the thirst level. From the first to the second level, P shows the largest effect, which is also the most significant effect. However, from the second level to the third, G shows more contribution. It means the considerable effect of G on the case depth happens at the higher values of G. Overall, regarding the case depth changes, P is the most effective parameter and the next important parameters are T and G with close effects on dRM. Blue lines in Fig. 7a expose a dominant influence of P compare to T and G, because of a steeper line of P over the three levels. Case depth changes over the three levels are more vivid in the edge plane wherein P and T mainly affect the case depth, then G with a considerable difference. It worth mentioning that G plays the same role as in the middle plane; very low effect from the first to the second level (the representative line for this part is almost flat) and steeper from the second to the third level.

Steep red lines of dTM, presented in Fig. 7b, emphasize the most influential rule of P and T on dTM. Almost the same length and steep line and the same starting and finishing point of P and T in their levels approve this deduction. This is while the influence of G on dTM is not comparable to P and T. Blue representative lines of dTE show that the effect of T is the most in all three levels while P is also considered as having the main effect on the case depth. In contrast, G has a minor effect compared to P and T.

3.3 Predictive models based on linear regression

To explain a mathematical relationship between the dependent and independent variables in a predictive model, regression equations based on minimizing mean squared errors, are exerted. The regression equations estimate responses (dRM, dRE, dTM, and dTE) depending on the parameters of this study (P, T, and G). As presented in section 3.1, the highly effective parameters concerning each response are analyzed by ANOVA and presented in Tables 6, 7, 8, and 9. The regression equations based on these analyzed parameters are presented for dRM, dRE, dTM, and dTE respectively in Eqs. 1 to 4.

$$ {d}_{RM}=-0.722+0.1042\times G+0.004321\times P+1.000\times T $$
(1)
$$ {d}_{RE}=-2.572+0.1458\times G+0.01111\times P+2.333\times T $$
(2)
$$ {d}_{TM}=-4.825+0.396\times G+0.02963\times P+6.667\times T $$
(3)
$$ {d}_{TE}=-5.905+0.02840\times \mathrm{P}+8.05\times T $$
(4)

The coefficient (denoted by R2 or R-squared) is an indicator to evaluate the precision of regression models. The R2 assesses the reliability of regression equations in the range of test variation. The R2 for Eq. 1 is calculated 91.67%, which declares that the equation can predict 91.67% of the variation case depth in dRM. The R2 for predictive equations of dRE, dTM, and dTE (Eq. 2 to 3) are 94.62%, 96.84%, and 93.42% respectively. These R2 values show that the regression equations are well fitted to experimental results. The graphical presentations of these predictive models (predicted), experiment results (measured), and the difference between (residual) versus case depths (dRM, dRE, dTM, and dTE) are demonstrated in Figs. 8 and 9. These curves present a visual comparison between predictive model and experimentation results. To evaluate quantitatively the preciseness of predictive models, for each of the four responses (dRM, dRE, dTM, and dTE), obtained case depths of models are compared to their experimentation outcomes. To this end, Eq. 5 is utilized to quantify statistically the error percentage of models.

$$ \mathrm{Error}\ \left(\%\right)=\frac{\left|\ \mathrm{Predicted}-\mathrm{Measured}\kern0.5em \right|}{\mathrm{Measured}}\times 100 $$
(5)
Fig. 8
figure 8

Case depth measured by experiments, predicted by regression equations and their residuals in a dRM b dRE

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Fig. 9
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Case depth measured by experiments, predicted by regression equations and their residuals in a dTM b dTE

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The average estimated errors for the response models (dRM, dRE, dTM, and dTE) are respectively 3.65%, 9.54%, 2.00%, and 8.03%. Results imply that case depths dRM, dRE, dTM, and dTE are computed with a good accuracy wherein the average estimation errors are less than 10%.

3.4 Response surfaces and sensitivity analysis

To investigate the trend of predictive response models (dRM, dRE, dTM, and dTE) with respect to their most effective terms, contour plots are drawn in Figs. 10 and 11 using MATLABTM. Contour plots are color-based maps drawn through two independent variables, presented on vertical and horizontal axes, displaying the variation a response by color gradient and isolines. These plots are useful when the objective is to specify which level combinations of parameters optimize the response. considering that the most important factors in our experimental responses are machine power (P) and heating time (T), the trend of case depth versus the variation of these parameters are presented in this study. The contour plot of dRM is presented in Fig. 10a where a linear relationship between machine power and heating time versus case depth is illustrated, which represents graphically Eq. 1 for dRM. It can be inferred from this plot that increasing P and T results in a deeper case depth in middle plane at tooth root of gear. In this regard, the maximum case depth can be obtained at the highest level of P and T. In Fig. 10b, the linear relation between independent variables (P and T) and response (dRE) is found similar to that of dRM. Increasing P and T leads to increasing the case depth. Thus, the highest case depth in dRE is also obtainable with the highest level of P and T.

Fig. 10
figure 10

Contour plots of P and T versus case depths at gear’s root in a middle plane (dRM) b edge plane (dRE)

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Fig. 11
figure 11

Contour plots of P and T versus case depths at gear’s tip in a middle plane (dTM,) b edge plane (dTE)

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Figure 11a presents the contour plot of P and T versus the case depth at gears’ tooth in its middle plane. It demonstrates more intense effects of P and T on case depth compare to their effect at the gear’s root, which is due to the proximity of tooth tip to the coil. This is concluded from a comparison of the highest dTM (3.8) which is shown in yellow color in Fig. 11a. The effects of changing P and T on dTE are displayed in Fig. 11b in which dTE raises with increasing P and T. Highest dTE could be achieved at the highest P and T. It is worth noting that the maximum case depth of dTE is slightly less than dTM in the range of our experimentations. The general conclusion in this section is that the case depth increases with the increase of P and T.

3.5 Edge effect discussion

The more radial distance of the coil from tip of gear results in a nonuniform case depth (as previously shown in Fig. 2c), where the case depth on the tip is more pronounced compared with the root of gear teeth. Referring to previous studies, induced eddy currents affect the case depth profile causing overheating of some areas, depending on the geometry of part which is exposed to induction heating. This leads to a case depth profile in edge plane that is different from the one in middle plane. The main purpose of this study is to investigate, analytically and experimentally, the machine parameters in order to minimize the edge effect, case depth difference in edge plane (dRE and dRM) over middle plane (dTE and dTM), using flux concentrators. To this end, the less case depth differences between middle and edge planes consider as a more ideal case depth profile in spur gears. Studying the effect of machine parameters on the case depth profile, monitoring it separately in middle and edge planes, it is concluded that increasing P and T results in a deeper case depth. Although reaching deeper case depth fulfills our goal to reach maximum induction heating performance, an ideal case depth profile is the one with the least edge effect. To measure the edge effect quantitatively, we introduce edge effect indicators of gears tooth at root, the difference of case depths measured in middle plain and edge of tooth root (ΔdR), and tip, the difference of case depths measured in middle plain and edge of tooth tip (ΔdT). The case depth difference between middle and edge planes at root and tip is shown in Fig. 12.

Fig. 12
figure 12

Edge effect comparison of experimental tests at root (∆dR) and tip (∆dT) of gears’ tooth

Full size image

The red line in Fig. 12 represents edge effect at tip of gears tooth. It shows a general enhancement in edge effect at tooth tip with respect to tests from 1 to 9. However, the increasing trend is not uniform, and a considerable reduction in edge effect happens in tests 3, 5, 7, and 9. This is because of an increase of both T and P in these tests compared to the others with an exception for test number 7 that a significant enhancement in T (0.1 s) compensate for P reduction from 170 to 143 kW. The minimum edge effect, and consequently the best case profiles at tip are achieved in tests number 1 and 3, where parameters P and T are at their minimum and maximum level respectively; 143 kW and 0.45 s for test number 1, and 170 kW and 0.55 s for test number 3. It should be mentioned that the effect of G is negligible compare to T and P due to their higher contribution percentage (Tables 8 and 9).

The blue line in Fig. 12 represents the edge effect at root of gears’ tooth. It shows a different trend compared to the edge effect at tip. The edge effect is less pronounced in tooth root compare to the tip since the case depth is generally less than tip due to its proximity to the induction coil. In this regard, the effect of G is not negligible at root (the contribution effect of G is 20.83% in middle plane and 8.15% in edge plane). From tests 1 to 9, G increased from 0.2 to 1 mm which shows a general decreasing trend in edge effect. The minimum edge effects happen at test numbers 3, 5, 6, and 9. However, test number 3 is the one in which the edge effect is minimum for both tip and root and provides higher case depth. Therefore, we can conclude that the optimum case depth profile is obtainable in test 3 wherein P and T are at their maximum level, 170 kW, and 0.55 s respectively.

4 Conclusions

In this article, 4340 steel-made spur gears are used to investigate the effects of induction heating process parameters namely machine power, axial gap between flux concentrators and the master gear, and heating time as independent variables (inputs) on the edge effect. The case depth is measured in four different areas on each gear: at tip and root of gears in edge plane and middle plane. Using Taguchi method, nine experiments are designed to combine four mentioned process parameters in three levels each. Measuring the case depths, the results are statically studied by analyses of variance (ANOVA). The predicted regression model of case depth in four areas of gears, at gear tips (dTM and dTE) and roots (dRM, dRE) are developed. The main effect plots, regression equations, error residuals, and contour plots of each response are developed and analyzed in this study. The results indicate that the total accuracy of predictive models is fine since the average errors of models are in an acceptable range: 3.65%, 9.54%, 2.00%, and 8.03% for dRM, dRE, dTM, and dTE respectively. Results demonstrate that power and heating time are the dominant parameters in case depth. The contribution percentage of machine power are 40.83%, 53.88%, 52.32%, and 41.91%, and of heating time are 30.00%, 32.59%, 36.33%, and 51.51% for dRM, dRE, dTM, and dTE in turn. Generally, the axial gap does not show a significant effect on the case depth. It does not affect the case depth in low levels, while it shows moderate effects in higher levels of the gap. The overall conclusion determines that case depth profile in middle plane is controllable via machine parameters, and the best-exploited range of parameters can maximize case depth while minimizing edge effect. The highest machine power and longest heating time in conjunction with the smallest axial gap contribute to an optimized case depth.

Although the results of this study are promising, some improvements and applications can be foreseen. As already mentions, this work can be improved by taking into consideration the effect of the radial gap of flux concentrators concerning the case depth of induction-heated gears. To optimize surface treatment of the other parts like discs, bevel gears, and helical gears which have different shapes than spur gears, experimental test benches as well as predictive models can be developed. In this regard, applying artificial intelligence to create and enhance the models on helical gear could be the next project to perform.