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Anewcase-depthestimationtechniqueforinduction-hardenedplatesbasedondynamicresponsestudiesusinglaserDopplerVibrometer

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online 5 September 2014 publishedProceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering

Mohan K Misra, Bishakh Bhattacharya, Onkar Singh and A Chatterjeestudies using laser Doppler vibrometer

A new case-depth estimation technique for induction-hardened plates based on dynamic response

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A new case-depth est imat iontechnique for induct ion-hardenedplates based on dynamic responsestudies using laser Doppler vibrometer

Mohan K Misra1, Bishakh Bhat tacharya2, Onkar Singh3

and A Chat ter jee2

Abst ractThis article presents the effect of case depth due to surface hardening on the dynamic response of steel alloy–based cir-cular plate. It indicates that such changes significantly affect the high-frequency response, especially, resonating modesand corresponding damping ratios of the plates. As the circular steel plates are quite rigid and hence the frequencyresponses are very small, a high precision three-dimensional laser Doppler vibrometer with a minimum velocity mea-surement capacity of 0.01m/s is used to obtain the modal response. A statistical approach based on response surfacemethodology is then applied to obtain the correlation between the dynamic response of the system and the effectivecase depth. Also, another sensitive parameter termed as curvature change factor is introduced for the determination ofthe hardened layer thickness on the basis of laser Doppler vibrometer data. Thus, anew technique of condition monitor-ingof case-depth profile of steel plates based on non-contact dynamic response of the system has been established.

KeywordsInduction hardening, case-depth profile, dynamic response, three-dimensional laser Doppler vibrometer, curvaturechange factor, response surface methodology

Date received: 9 April 2014; accepted: 24 July 2014

Int roduct ion

Surface/through hardening is most essential for improv-ing wear resistance and strength of steel components. Itis achieved when a selected surface or the entire objectis subjected to a cycle of heating and quenching.1

However, the hardening process requires careful moni-toring of the system parameters to obtain a controlledcase-depth profile which is generally carried out usinginduction hardening process.

Electromagnetic induction hardening is an energy-efficient, in-line, heat-treatment process widely used inautomotive industry to surface-harden automotiveparts at the lowest possible cost. The substitution ofinduction hardening for furnace hardening may leadto savings of up to 95% of the energy used in theheat-treating operations.2 Also, such process causes sig-nificant weight-saving in automotive power-train com-ponents, further saving energy and manufacturing cost.Proper control of induction hardening process hingeson appropriate selection of coil design, selection of

excitation frequency, and timing of heating andquenching. This process is generally based on trial anderror which incurs loss of material and time. To mini-mize the effect of these errors on the quality of induc-tion, hardened sample is perceived to be a difficult taskdue to enormous system complexity.

It is generally observed that induction hardeningprocess enhances the strength and wear resistance ofthe surface layer of the components at the cost of

1PhD Scholar, U PTechnical University, Lucknow, India2Department of Mechanical Engineering, Indian Institute of Technology,Kanpur, India

3Department of Mechanical Engineering, Harcourt Butler TechnologicalInstitute, Kanpur, India

Corresponding authors:Mohan K Misra, SMSSLab, NL#102,I I T Kanpur- 208 016 India.Email: emkay@iitk.ac.inBishakh Bhattacharya, Department of Mechanical Engineering, IndianInstitute of Technology, Kanpur- 208 016, IndiaEmail: bishakh@iitk.ac.in

ductility and toughness.2 Therefore, it is useful to esti-mate the mechanical properties of hardened samplesbased on the micro-structural condition at the subsur-face region of a hardened component. As it is practi-cally impossible to monitor the micro-structuredirectly, hence, it is imperative to devise a non-destructive evaluation (NDE) technique which cansense the change in system parameters due to finermicro-structure variations. The major advantage ofusing non-destructive techniques is to provide an eco-nomic way of achieving high-quality inspection whichis not possible with the destructive testing methodsused to determine the hardened layer profile of shaftsand other components. The methods commonlyemployed for measuring such case depth are visual(with an acid etch), chemical, mechanical, and non-destructive techniques.3

In this article, a brief overview of various non-destructive techniques, generally used for the measure-ment of case-depth profile, is presented. Next sectiondiscusses the use of a new dynamic response–basedtechnique for this purpose. Modal parameters alongwith curvature change factor (CCF) have been shownto be effective in this direction. Subsequently, based onthe dynamic response, a predictive model for effectivecase depth (ECD) has been developed. The results arefurther validated with the help of destructive studies.These are carried out for selected plates with variedcase depths. It has been shown to be effective in pre-dicting the case-depth profile accurately. F inally, con-clusions are drawn based on the study.

An overview of case-depth measurementtechniques

Before discussing on NDEs of hardness and casedepth, it is important to distinguish between ECDand total case depth of any hardened steel sample.ECD or the thickness of the hardened layer is anessential quality parameter of the induction

hardening process which is defined by the user basedon application. The darker periphery of a typicalround plate, as shown in F igure 1(a), shows the ECDof a hardened sample.

For a typical round plate, ECD is determined as thatregion or distance from periphery where desired hard-ness value is measured. In our case, it is 58-60 HRC forcomponents like spindle shaft. F igure 1(b) shows ECDas typically 65% of the total micro-structure case depth.Micro-structure case depth is considered to be 75% ofthe total case depth which is also defined as the distancefrom periphery where hardness value is 10% more thanthe hardness at the core of the steel sample.4

To ascertain hardness and case depth, NDE tech-niques are used based on elastic, electrical, and mag-netic material properties. In the case of ferrousmaterials, often magneto elastic parameters are alsoused. These properties may vary, depending on micro-structure, macro-stresses, micro-stresses, anisotropy,and further intrinsic properties.5,6

A review of various NDE-based case-depth detectionprocedures brings out following major techniques thatare commonly employed:

1. Use of the magnetic and magneto-electricalcharacteristics;7

2. Sound velocity of ultrasonic waves;8,9

3. Backscatter of ultrasonic waves;10

4. Other techniques, for example, potential dropmeasurement.11

These procedures are briefly described in thefollowing:

1. In this technique, the magnetic Barkhausen emis-sion (MBE) from the hardened profile of carbonsteel is compared with non-hardened samples andthe extent of surface hardening is predicted.12

Generally, the hysteresis loop shows a distortionwith a sudden reduction in the rate of

Figure 1. (a) EN-8 sample with ECD of 4mm and (b) a typical profile of asurface-hardened EN-8 sample.

2 Proc IMechEPart I: JSystemsand Control Engineering

magnetization (dB/dH) indicating the presence ofa distinctive boundary of hardened surface. Thesystematic changes in the MBE profile for differentvoltages applied during induction heating indicatethe dynamic changes in micro-structural configura-tions. However, this process is effective in measur-ing low case depth only. The limiting factor forMBE is the attenuation of the signal source byeddy currents as it propagates through the mate-rial, leading to a maximum measurement depth ofabout 1 mm,5,13 depending on the excitation fre-quency, frequency range, and material properties.The domain wall motion that generates MBE alsocauses a release of elastic energy known asmagneto-acoustic emission (MAE), which has agreater measurement depth, and hence offers acomplementary technique to extend the measure-ment depth up to 8 mm for the characterization ofsurface treatments in steel.12

2. From Figure 1, it is evident that the change inmicro-structure generates several interfaces in ahardened steel beam and thus causes sound wavesto get reflected at the interfaces which form thebasis of ultrasonic testing (UST). Theiner et al.14

have performed ultrasonic tests, in which soundwaves of high frequency (10–25 MHz) are trans-mitted into the steel sample (rod placed in a watertank). The transducers have sensed the boundarybetween soft core and hardened layer as disconti-nuities and therefore predicted the depth ofhardened layer as a function of time delay or time-of-flight (TOF) between the front surface reflectionand the reflection at the transition zone.6,14 This

procedure is further modified to developultrasonic-based backscattering technique.

3. In ultrasonic backscatter technique (USBT), thespecimen and ultrasonic probe are both immersedin a tank filled with water (see Figure 2). The place-ment of probe is made in such a way that it cansend shear waves into the specimen; it is inclined atan angle from the orthogonal cross-sectional planeof the specimen.15 Ultrasonic backscattering phe-nomenon occurred twice, the first echo takes placeat the surface of the specimen due to surface rough-ness and results in the return of part of the energyto the probe. Ultrasonic energy also enters the har-dened layer as a shear wave. The hardened surfacelayer is made of fine martensite structure and thus,no scatter of ultrasonic wave takes place in thisregion. However, when the shear wave reaches thetransient zone (TZ), where martensite structure isgradually converted to ferrite-pearlite structurewhich is of a larger grain size, once again, theenergy is scattered at the grain boundaries. Thistransition zone backscatter forms the second echo,as shown in Figure 2.

4. In alternating current potential drop (ACPD) tech-nique, the standard values of conductivity and per-meability of an unhardened rod are used tocalibrate the experimental results of induction-hardened steel rods.11 The hardened rod is assumedto be made up of a homogeneous soft core which issurrounded by a case hardened layer of uniformthickness. The values of case depth are obtained byparameter fitting to minimize a penalty functionusing multi frequency potential drop measure-ments. This method has shown a reasonable agree-ment with hardness values found using Rockwellhardness measurements.

A comparison of different traditional NDE pro-cesses used for surface profile estimation has beenshown in Table 1. Table 1 shows that ultrasonic andMBE-based techniques are highly accurate. However,these are very expensive and difficult to be carried outin real time.

The study of dynamic response of a system couldprovide very significant indications toward the changeof micro-structure or phase change of a material.16 Thenext section presents a brief description of the currentuses of dynamic response in terms of damage detectionof various mechanical components. Subsequently, thistechnique is extended for case-depth prediction.

Dynamic response for systemident ificat ion

The use of dynamic response in structural health moni-toring is getting popular nowadays. The changes indynamic responses such as natural frequencies, modaldamping, and mode shapes of the structures are

Figure 2. Scatter of ultrasonic waves from the surface andtransient zone of an induction-hardened specimen.

Misra et al. 3

observed when changes in physical properties of thestructures occur due to damage. Therefore, throughsupervised training, this knowledge-base of dynamicresponse proves helpful for damage identification.Measuring changes in natural frequency to detect dam-age often face limitation, as these are mostly coupledwith inherent noise, especially in large structures.Sometimes very small changes in natural frequency(which is registered as noise) attribute due to tempera-ture difference and change of mass and hence may gen-erate false results.16 The location of damage in flatplate based on finite element analysis is discussed byKumar et al.17 They have shown significant changes inflexural vibration modes for flat plate while assumingmodulus of elasticity in the damaged zone to beclose to zero. Hanagud et al.18 have noted the charac-teristics of lower-order curvature modes, which arefound to be distinctive for healthy and damaged struc-tures. Thereafter, a mathematical relationship for thedynamic parameter (difference in amplitude of curva-ture mode shapes) is employed to predict the loss oflocal stiffness which was caused as a result of damage.For small change in natural frequency, the frequency-based damage detection methods are ineffective in pre-dicting the extent and location of damage. Luo andHanagud19 have suggested a detection algorithm whichis based on integral change in curvature mode shape ofa structural beam and is found to be a better index fordamage detection.

Thus, the literature survey indicates that the dynamicresponse of a system in various forms, namely, displace-ment, velocity, natural frequency, damping, modal fre-quencies, and mode shapes, has been used extensively fordamage detection. As surface hardening is expected tocause local changes (comparable to damages in structures)in the disk structure, hence, similar techniques could alsobe employed to detect the hardening profile. The followingsection provides a brief description of the changes in steelstructure that occurs during surface hardening and how itcan affect the mechanical property of the plate.

Dynamic response of hardened plate

Medved and Bryukhanov20 have shown that in accor-dance with Born’s theory, the reduction of Young’smodulus is inevitable when distortion of the crystal lat-tice occurs in a crystalline system as a result of any

hardening process. It is well known that medium car-bon steel (C: 0.45% ) while undergoing induction hard-ening process, at temperatures above 730 °C,transforms into austenite phase. As soon as the steelcomponents slowly get cooled, the austenite phasechanges into ferrite structure. The dynamic response ofhardened steel plate is expected to respond to suchphase transformation. In the current study, modalanalysis of EN-8 samples is carried out using laserDoppler vibrometer (LDV), and the vibration signatureas well as damping analysis are carried out for unhar-dened and hardened samples in order to obtain a corre-lation between thickness of the hardening layer andsystem response. Figure 3(a) and (b) shows typical fre-quency response, in pre and post hardening stage,explaining this concept very well.

It is evident that a part of excitation signals, gener-ated from shaker, traveled through hardened surfacewhere some part is reflected from interface between softand hardened layers (reflected signal shown in red inFigure 3(b). Therefore, the resultant wave signal (madeup of refracted as well as reflected wave) produceschange in natural frequency v n and damping ratio z.This change in dynamic response will be different forvarying case depths and thus may prove beneficial asan important indicator for predicting ECD.

It may be noted from Figure 4 that a hardened speci-men shows significant change in the natural frequencyin comparison to the unhardened sample. Applicabilityof experimental modal analysis for predicting in-processcase depth is discussed elaborately in the followingsections.

Brief description of the polytec scanning vibrometer(PSV) three-dimensional measurement method

In laser-based scanning, the scanner sends a low-powerlaser light on the surface of the vibrating object (seeFigure 5), following a grid/mesh which has been givenas an input to the system.

A charge-coupled device (CCD) camera integratedwith the scanner records the position coordinates andlight intensity at each point on the surface of the object.Taking recourse of the Doppler effect, these data aretransformed to obtain the velocity response of everyscanned point. The velocity response profile is thendigitally integrated to obtain the deflection response.

Table 1. Comparison of different traditional NDEprocesses.

Method Phenomenon Range of measurement Accuracy (%)

MBEand MAE12 Electro-magnetism 7–8mm ~1.5%UST14 Reflection of sound wave 2mm Less than 5%USBT15 Reflection of sound wave 2–4mm ~0.5%ACPD11 Change of conductivity and permeability 2mm 20%–24%

MBE: magnetic Barkhausen emission; MAE: magneto-acoustic emission; UST: ultrasonic testing; USBT: ultrasonic backscatter technique; ACPD:alternating current potential drop.

4 Proc IMechEPart I: JSystemsand Control Engineering

This procedure is carried out for all the predefinedpoints on the surface of the object to get the completeoperational deflection shape (ODS) of the object underdynamic loading. The three-dimensional (3D) LDV is alaser-based non-contact vibration measuring instru-ment which consists of three measuring scan headscapable of measuring the velocity and displacementalong any three specified directions yielding full infor-mation of the 3D ODS. The minimum detectable

vibration speed using this system is 0.01 m/s (at 1 Hzresolution), while the maximum speed is of 10 m/s. Afunction generator (NI-671x) contained in the system isused for generating excitation signals in the frequencyrange of 0–80 kHz. The LDV can measure vibrationsup to 30 MHz range with very linear phase responseand high accuracy. For the dynamic excitation of theEN-8 sample, an electro-dynamic shaker is used. Apower amplifier (make: LDS, PA500L series) is con-nected to the shaker for the purpose of amplifying theexcitation signal generated by the LDV system. Duringthe experiment, a pseudo random signal in the fre-quency range of 0–7500 Hz is used for the excitation ofthe samples (Table 2).

Measurement of damping

A quantitative estimation of damping ratio z isobtained using the well-known half-power bandwidthmethod, as shown graphically in Figure 6. The damp-ing ratio, z, can be determined using the followingrelationship

z =Dv2v p

ð1Þ

Figure 4. Change in dynamic response due to hardeningofsteel sample.

Figure 5. A schematic diagram of the experimental setup.LDV: laser Doppler vibrometer.

Figure 3. (a) Dynamic response of unhardened component and (b) dynamic response of hardened sample.

Misra et al. 5

Dv is determined from the half-power points (v 1 andv 2), as shown in Figure 6.

The loss factor, h, can be obtained from strainenergy density using the following relationship

h =DU

2pUmaxð2aÞ

h ’ 23 z ð2bÞ

where Umax is the maximum strain energy stored in thesystem and DU is damping capacity for the sample disk.

Exper imentat ion on dynamic responsestudy

Sample preparation

EN-8 steel samples with diameter of 38 mm and thick-ness of 32 mm are hardened under varying processparameters on Inductotherm make 250 kW, 3–10 kHz

induction heater model UP-12. Table 3 shows processparameters which produced samples with ECD of 4, 7,and 12 mm.

In order to achieve high-amplitude vibrationresponse from experimental EN-8 samples using LDV,a very thin slice of 2.5–2.6 mm was cut with copperelectrode in electrode discharge machine (EDM). TheEDM was used to avoid apparent changes in casedepth or hardness of the samples. Steel samples werethen coated with developer so that the laser signalscould easily get reflected from the surface to achieveaccurate results. Experiments are carried out with anincreased amplifier gain of 2.5, in order to have suffi-cient excitation of the high-frequency modes. Timerequired for the complete scan depends on the numberof scan points defined and fast fourier transform (FFT)parameters. In the present analysis, only 25 scan pointswere defined due to smaller size of EN-8 sample (seeFigure 7(c)). F igure 7(a)–(c) shows the overall testsetup, the disk mounted on the shaker and scanningpoint representation, respectively.

Dynamic response results for case-depth predictionusingresponse surfacemethodology

Response surface methodology (RSM) is a useful toolfor predicting the behavior of response characteristicsfor any process especially manufacturing. It helps toanalyze the effect of input parameters on output usingdifferent statistical and mathematical methods.

Following are the requisites to utilize RSM for pre-dicting response characteristics of a process:

1. Data collection for those independent variableswhich influence the system greatly;

2. Prediction of suitable model of the system forresults based on the experiments.

In this pursuit, Abou-El-Hossein et al.21 predictedthe cutting force in end-milling operation of tool steel,using the cutting force equations developed with thehelp of RSM. The effects of all the input parameters,for example, cutting speed, fed rate, radial depth, andaxial depth of cut, were studied to develop predictivemodel for cutting force. Similar literatures are available,where RSM is successfully used for predictive modeling

Table 2. Details of the excitation signal used in 3D scan mode.

Excitation signal Pseudo random

Frequency range 0–7500HzFFT lines 1600Window RectangleAverages 10 (complex)Velocity decoder VD-09 digital decoder (velocity

range: 0–10mm/s)

Figure 6. Measuringdamping using3-dB method.

Table 3. Process parameters to obtain output with desired ECD.

Number Process parameter Value to obtain 4-mm ECD Value to obtain 7-mm ECD Value to obtain 12-mm ECD

1 Frequency 10kHz 2kHz 2kHz2 Power 65kW 65kW 65kW3 Current 105A at 415V 105A at 415V 105A at 415V4 Scanning speed 1300mm/min 980mm/min 550mm/min5 Quench flow 40L/min at 22 lbf/in2 40L/min at 22 lbf/in2 40L/min at 22 lbf/in2

6 Quench temperature 32°C 32°C 32°C7 Surface hardness 58–60HRC 58–60HRC 58–60HRC

ECD: effective case depth.

6 Proc IMechEPart I: JSystemsand Control Engineering

of response characteristics of wide range of industrialprocesses.

Our approach is to first build a model based onknown ECD samples and then validate this modelthrough destructive testing. Therefore, the suggestedmodel would be helpful for predicting the case-depthprofile of output samples for which we only know thedynamic response. In our present study, 48 EN-8 har-dened samples are taken. These hardened samples areessentially the output of induction hardening processbased on parameters presented in Table 3. This is doneto ensure a robust model for predicting ECD in concur-rence with dynamic response parameters. With the helpof LDV, output graph for velocity magnitude in deci-bels and frequency response of all the samples areobtained. Once analyzing the results, dynamic responseparameters such as natural frequency, damping ratio,and loss factor are identified. These parameters areused as input variables for Design-Expert software forpredicting the response factor which is case depth inour case. This is a statistical software package whichuses design of experiment (DOE) methods for optimi-zation of process. The design summary, presented inTable 4, is a brief description of input data and pro-posed analysis method provided to the software forcase-depth predictive model using RSM. Here, designmodel refers to the choice of model order and nature ofresponse. It is used for simulating the nature of theinput–output relationship on the basis of supplied datato the software, which is the information gathered from48 EN-8 samples. Since the variation in minimum andmaximum values of damping ratio is very small, stan-dard deviation value for damping ratio did not appearin this presented design summary.

Result s and discussion

Table 7 presented in Appendix 1 shows the experimen-tal results corresponding to 16 hardened samples with

ECD of 4, 7, and 12 mm each, which are output ofinduction hardening process based on parameters men-tioned in Table 3. Therefore, samples are classifiedbased on their known ECD values. Response plots andanimated mode shapes were studied after the scan toidentify the nature of the modes and match betweenthe similar modes of different samples.

F igure 8 shows the rigid body mode and first bend-ing mode for an unhardened plate while subjected tobase excitation. The natural frequency obtained, for thefirst bending mode, is 4691 Hz and the correspondingdamping factor is 0.024.

The dynamic response of hardened plate with 12 mmcase depth as shown in Figure 9 appeared shifted, andnatural frequency corresponding to the first bendingmode is obtained as 4568 Hz. A similar change in natu-ral frequency of the first bending mode for hardenedplate with 4 and 7 mm case depths has also beenobserved. A drop of 2% –2.5% in natural frequency isobserved due to hardening of the steel plate.

This observation is indicative in terms of predictingthe case depth obtained as a result of induction harden-ing procedure. In order to develop a predictive modelfor case depth using dynamic response results, data pre-sented in Table 7 (see Appendix 1) are used as historicaldata under response surface environment with Design-Expert software.

F igures 10 and 11 show the scatter plot related tonatural frequency and damping ratio, respectively, forsamples with different case depths. It may be noted thatwhile the damping ratio has not varied much from sam-ple to sample, the natural frequency shows variation ofabout 0.5% from the mean value. This is possible dueto variation of ECD during induction hardeningprocess.

F igure 12 shows scatter plot between natural fre-quency and damping ratio. Also from this figure, it canbe concluded that damping ratio increases with increasein case depth. This observation is supported well in

Figure 7. (a) PSV 3D experimental setup, (b) EN-8 sample mounted on electro-dynamic shaker, and (c) sample with scanningpointrepresentation.

Misra et al. 7

findings of Visnapuu et al.22 where it is discussed thatheat treatment enhances inter-granular boundary stresslevel and thus improves damping. Possibly, this is thereason that with increase of case depth, damping ratioincreases.

For the presented historical data, various combina-tions of process orders (i.e. Cubic, Quadratic, Mean,2F1, linear, and so on) have been tested for obtainingthe significance of process parameters and lack of fit.This significance test is needed to avoid any error ornoise for the predicted output. Table 5 shows the analy-sis of variance (ANOVA), for concluding the bestmodel for case-depth prediction. For linear processorder, F value is found to be quite high. The high valueof F suggests that there are very little chances(\ 0.01% ) of any error or noise. Therefore, reducedlinear model for case-depth predictions proved as sig-nificant. Also, the absence of loss factor suggests theinsignificance of loss factor parameter in the presentedset of data for predicting the ECD.

Thus, ANOVA concluded the predicted model forcase depth which is described in equation (3)

Predicted ECD = �55:718 + 0:00943 v n + 7512:773 z ð3Þ

In order to establish the efficacy of this dynamicresponse–based innovative NDE technique, destructivetesting of some of the experimental samples was carriedout. Following section presents the validation of modelfor case-depth prediction as well as results obtainedfrom micro-structure.

Validation of dynamic response technique

To establish the accuracy of results obtained fromdynamic response–based predictive model, 6 samplesout of 48 experimental samples were selected for furtherdetailed testing. Micro-hardness and micro-structuresprovide definite information about the ECD of theseexperimental samples. F igure 13 shows the micro-structures of one of these samples which were obtainedfrom metallurgical microscope model: RMM-2 usingProgRes CT3 camera.

The micro-structure is observed, at different placesof the EN-8 sample, to ascertain the actual phases ofsteel. It is very clearly visible under the microscope atdifferent resolutions (4003 , 10003 ) that the middlelayer of sample is having a mixture of ferrite and pear-lite. The outer hard layer is pure martensite (needle-likestructure).

The standard testing procedure for Rockwell hard-ness test method was adopted, and samples were inves-tigated for average value of micro-hardness at differentregions on those experimental samples (Figure 14).These regions are approximately 4 mm wide zone. Ithelps us to locate the steep drop of hardness value sothat ECD can be identified. Table 6 shows the resultswhich validate the case-depth prediction model verywell.

CCF

While change in modal parameters may be directlylinked to the system hardness, the determination of

Table 4. Design summary for RSM for case-depth predictive model.

Study type Response surface

Initial design Historical dataDesign model QuadraticRuns 48Factor Units Low actual High actual Mean Standard

divisionNatural frequency (v n) Hz 4558 4610 4582 15.808Damping ratio (z) 0.00217 0.00329 0.003 –Loss factor (h) 0.00434 0.0658 0.005 0.001Response Units Analysis Minimum Maximum Mean Standard

divisionMethod

Case depth mm Polynomial 4 12 7.680 3.297 Reduced linear

Figure 8. Natural frequency v n obtained from unhardenedsample.

8 Proc IMechEPart I: JSystemsand Control Engineering

hardness profile may need further analysis of thedynamic response. In case of health monitoring, dam-age detection algorithms are capable of extracting

useful information related to indication of damagelocation and magnitude. A similar index is developedhere for obtaining the hardness profile by curvaturechange as an efficient parameter. The post-processingdata obtained from LDV measurement give coordi-nates and velocity magnitude (m/s) for all the nodes (inour case there are 25 nodes) at v n which is fundamentalfrequency corresponding to the bending mode. Forevery node, velocity, displacement, and mode shape areobtained from the LDV. The mode shape curvature(MSC) may be calculated from the measured displace-ment of mode shapes using a central difference approx-imation as

€f ji =f (j + 1)i � 2f ji + f (j�1)i

L2ð4Þ

where €f is the mode shape corresponding to the trans-verse vibration, i is the mode shape number, j is thenode number, and L is the distance between the nodesand represents spatial double differentiation.

Effect of hardening on EN-8 samples of various casedepths can be more accurately predicted with the helpof MSC. In case of induction hardening process, whena hardening layer is formed, the transverse stiffness isreduced while the magnitude of modal curvatureincreases. The absolute difference between the modalcurvature of the unhardened and hardened samples ishighest in the interface and reduces toward the corewhich is stiffer. The MSC criterion is defined here asthe difference in absolute curvatures of the unhardenedand hardened samples for each mode and representedas

MSC = D€f i = €f Unhardened

� �j�

€f Hardened

� �j

���

���i

ð5Þ

The value of CCF is then calculated with the help ofcurvature values at the same nodes of unhardened andhardened steel samples considering all the mode shapes.Hence, it is the true comparison of dynamic responsevalue if any change in micro-structure occurs in steelsample due to induction hardening process. In otherwords, change in case depth would lead to change in

Figure 9. Natural frequency v n obtained from hardenedsample with 12mm case depth.

4 mm7 mm

12 mm

4550

4560

4570

4580

4590

4600

4610

4620

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

Nat

ural

Fre

quen

cy (

Hz)

Figure 10. Natural frequency distribution across differentcase-depth samples.

4 mm7 mm

12 mm

0.002

0.0022

0.0024

0.0026

0.0028

0.003

0.0032

0.0034

0.0036

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Sample Number

Dam

ping

Rat

io (ζ)

Figure 11. Damping ratio distribution across different case-depth samples.

4 mm7 mm

12 mm

0.002

0.0022

0.0024

0.0026

0.0028

0.003

0.0032

0.0034

4550 4560 4570 4580 4590 4600 4610

Dam

ping

Rat

io

Natural Frequency

Figure 12. Damping ratio and natural frequency relationshipacross different case-depth samples.

Misra et al. 9

natural frequency as well as damping ratio; therefore,there will be a definite change in MSC and CCF as well.However, since all modes may not be equally sensitiveto the hardness profile, the average change over a fixednumber of modes gives the CCF defined in equation(6), which is more reliable to predict the effect of hard-ening. Thus

CCF =1N

XN

i = 1

€f Unhardened

� �j�

€f Hardened

� �j

���

���i

ð6Þ

where N is total number of modes, i is the mode shapenumber, and j is the node number. Since in the currentstudy, only one bending mode is observed in the fre-quency range of 7500 Hz, therefore, the value of MSCand CCF remains identical. This developed index calledCCF is very useful in comparison to conventional

micro-hardness methods. It is also useful for curvedsurfaces.

For calculating CCF, experimental process can besummarized as per flow chart presented in Figure 15.As per ASTM designation E18-12 (standard testmethod for Rockwell hardness testing for metallicmaterials), the indentation spacing defined to be 3dbetween two indents while 2.5d spacing from the edge,where d is the diameter of the indenter (Figure 16).

For the curved surface, it is highly likely to predictthe case-depth profile with lesser accuracy, especially inzones where measuring hardness is not possible due tothis minimum spacing constraint. On the other hand,no such restriction is applicable for CCF which is morecontinuous and can show a regular contour of hardenedsurface of test sample. This distinction clearly estab-lishes the usefulness of this newly developed CCF index.

Table 5. ANOVA for response surface reduced linear model.

Source Sum of squares df Mean square Fvalue p-value (probability . F)

Model 521.66 2 260.83 54589.34 \ 0.0001A-natural frequency 0.12 1 0.12 25.36 \ 0.0001B-damping ratio 64.52 1 64.52 13503.37 \ 0.0001Residual 0.22 45 0.00478Lack of fit 0.20 27 0.00743 9.20 \ 0.0001Pure error 0.015 18 0.00081Correct Total Sum of Squares 1462.49 75

Figure 13. Micro-structure of unhardened and hardened layers of sample.

10 Proc IMechEPart I: JSystemsand Control Engineering

Figure 17 shows the CCF plot for a 4-mm case-depthhardened sample.

The green points correspond to the scanned nodesthat lie inside the less stiff yet hardened layer (Figure7(c)). The value of CCF is comparatively higher in thisregion. As we progress toward the core of the sample,the blue points appeared which is relatively stiff andnon-hardened. Correspondingly, we see lower values ofCCF in this region. Based on the study of a group ofsamples, a threshold value of approximately 8000 isfixed, which corresponds very well to the beginning ofthe unhardened layer.

Conclusion

In this article, a new dynamic response–based techniquefor the determination of induction hardening profile ofsteel samples is presented. To carry out this study, steeldisks with three different hardness depths of 4, 7, and12 mm were prepared. Metallurgical microscope imagesof the hardened and polished disks confirmed the pres-ence of a two-layered profile. The outer hardened layerwas martensite, which was followed by a ring of ferriteand pearlite below the hardened layer. The hardeninglayer was subsequently confirmed from the micro-hardness testing. Next, the disks were installed on ashaker in a cantilever mode, and the dynamic responseof the top surface was obtained using LDV. By analyz-ing the response data, a noted difference between theunhardened and hardened sample was observed corre-sponding to natural frequency, damping ratio, andmodal loss factor. With the help of available dynamic

response data for 48 EN-8 samples, a case-depth predic-tive model was developed using RSM technique withthe help of Design-Expert software. To further inte-grate the system parameters, a new metric named CCFis proposed which has been shown to have a strong cor-relation with the hardening profile.

Table 6. Results for validation of predicted model.

Sample number Harness atregion A

Hardness atregion B

Hardness atregion C

Hardness atregion D

Actual ECD PredictedECD

Absoluteerror

10 59 47 35 24 4 4.01 0.22%16 60 45 32 22 4 3.96 2.42%24 60 58 40 28 7 7.02 0.28%30 58 59 36 26 7 7.01 0.14%39 58 60 60 32 12 11.86 1.17%43 60 60 59 30 12 11.88 1.00%

ECD: effective case depth.

Figure 14. A schematic diagram of the experimental setup.

Figure 15. Flow chart for calculatingCCFfor steel samples.2D: two-dimensional; CCF: curvature change factor.

Misra et al. 11

In future, the CCF will be augmented with the otherdynamic response parameters to obtain a better predic-tion of ECD. Also, this dynamic response–based expertsystem will be directly integrated to a control systemfor the effective control of induction hardening processfor components of straight as well as curved surfaceprofile.

Acknowledgements

The authors gratefully acknowledge partial support ofUKIERI (Project: UKIERI/MDES/20120045) for dis-seminating the results in a national conference.

Declarat ion of conflict ing interests

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

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Misra et al. 13

Tab

le7.

Expe

rim

enta

lres

ults

for

hard

ened

stee

lsam

ples

..

Sam

ple

num

ber

Nat

ural

freq

uenc

y(v

n)

Dam

ping

ratio

(z)

Loss

fact

or(h

)EC

D(m

m)

Sam

ple

num

ber

Nat

ural

freq

uenc

y(v

n)

Dam

ping

ratio

(z)

Loss

fact

or(h

)EC

D(m

m)

145

982.

175E2

034.

350E2

034

2545

722.

625E2

035.

249E2

037

246

012.

173E2

034.

347E2

034

2645

742.

624E2

035.

247E2

037

346

042.

172E2

034.

344E2

034

2745

872.

616E2

035.

232E2

037

445

972.

175E2

034.

351E2

034

2845

782.

621E2

035.

242E2

037

545

922.

178E2

034.

355E2

034

2945

802.

620E2

035.

240E2

037

646

022.

173E2

034.

346E2

034

3045

782.

621E2

035.

242E2

037

745

982.

175E2

034.

350E2

034

3145

852.

617E2

035.

234E2

037

846

062.

171E2

034.

342E2

034

3245

742.

624E2

035.

247E2

037

946

022.

173E2

034.

346E2

034

3345

583.

291E2

036.

582E2

0312

1046

102.

169E2

034.

338E2

034

3445

613.

289E2

036.

577E2

0312

1146

082.

170E2

034.

340E2

034

3545

623.

288E2

036.

576E2

0312

1245

982.

175E2

034.

350E2

034

3645

703.

282E2

036.

565E2

0312

1345

992.

174E2

034.

349E2

034

3745

653.

286E2

036.

572E2

0312

1446

002.

174E2

034.

348E2

034

3845

683.

284E2

036.

567E2

0312

1545

922.

178E2

034.

355E2

034

3945

603.

289E2

036.

579E2

0312

1646

022.

173E2

034.

346E2

034

4045

643.

287E2

036.

573E2

0312

1745

882.

616E2

035.

231E2

037

4145

583.

291E2

036.

582E2

0312

1845

802.

620E2

035.

240E2

037

4245

683.

284E2

036.

567E2

0312

1945

782.

621E2

035.

242E2

037

4345

663.

285E2

036.

570E2

0312

2045

872.

616E2

035.

232E2

037

4445

583.

291E2

036.

582E2

0312

2145

742.

623E2

035.

247E2

037

4545

623.

288E2

036.

576E2

0312

2245

872.

616E2

035.

232E2

037

4645

703.

282E2

036.

565E2

0312

2345

882.

616E2

035.

231E2

037

4745

623.

288E2

036.

576E2

0312

2445

792.

621E2

035.

241E2

037

4845

683.

284E2

036.

567E2

0312

Ap

pen

dix

114 Proc IMechEPart I: JSystemsand Control Engineering