Applying acceleration and strain signals for the stress ... · important factor in the stress...

Applying acceleration and strain signals for thestress evaluation of a steel cutter

Konsta Karioja and Sulo Lahdelma

Mechatronics and Machine Diagnostics Laboratory, Department of MechanicalEngineering, P.O. Box 4200, FI-90014 University of Oulu, Finland

E-mail: [email protected], [email protected]

Abstract

Mechanical cutting is an old, effective method for steel cutting. The variety of steelsavailable in the market has changed considerably over the last few years, and newdemands for mechanical cutting have been set. The shearing force is of course a veryimportant factor in the stress evaluation of a cutter, but also vibrations should beconsidered. This paper discusses the evaluation of stress in steel cutters by meansof strain and acceleration signals. It also investigates the possibility of detectingdifferent types of stress by using real order derivatives and their weighted lp norms.The MIT measurement index is utilised in order to determine stress levels. It is alsodiscussed whether a strain gauge or an accelerometer alone is sufficient in this caseif signal processing is performed in a suitable way.

Keywords: steel cutter, acceleration, strain, stress evaluation, measurementindex, weighted lp norm, fractional derivative, real order derivative

1. Introduction

Different mechanical cutters have been widely used in the industry for decades. Itis not easy to determine the kinds of modern steels that can be cut with old cutters.This creates a need for the stress evaluation of cutters. Instead of mechanical cutting,plasma cutter, for example, can be utilised, though the method is considerably lesseffective. If mechanical cutting is not applied when it would be possible withoutrisk, some of the production capacity will be wasted.

In this case the target of the measurements was a guillotine type steel cutter. Thecutter is located at Ruukki Metals Raahe Works and is part of the machinery of therolling mill. The cutter has been in use for decades, so it can not be designed to cutmodern high strength steels. The materials of cutting blades are of course differentnowadays, but the cutter itself has only undergone minor modifications.

The Tenth International Conference on Condition Monitoring and Machinery Failure Prevention Technologies

2. Guillotine cutting

Guillotine cutting as a method has been the subject of some research in the past (1,2,3)

and there are also some handbooks (4,5) about steel cutting with different kinds ofmethods including guillotine cutting. Unfortunately, question about the stress ofthe cutter seems to have been under consideration very seldom.

The cutting process of guillotine cutting is sketched in Figure 1. The maximumshearing force needed is of course a very important aspect in the stress evaluationof cutters. The maximum force is proportional to plate thickness (t), which is quitea straightforward conclusion. On the other hand, the strength of the material is notdirectly proportional to the maximum cutting force (1). The strength of material,however, has an effect on shearing force, but a more crucial factor is the susceptibilityif the material to a brittle fracture. The explanation below, which refers to Figure1, clarifies this statement.

At the point A in Figure 1 the blade is on the upper surface of the plate, and in thepoint C the blade has reached the lower surface. The cutting is nonetheless takingplace only between points A and B, not between point A and point C as one mightassume. The area coloured grey is cut, and the blade has already reached it, and inthe white area the material is still intact. The red area is also cut, because a brittlefracture is caused by the shearing force, so the material is being cut only in the bluearea in Figure 1.

When the material is very hard and susceptible to a brittle fracture, the bladepenetration (y) will be considerably smaller than with conventional steels. Thismeans that distance x will also be reduced. Thus the blade will be in contact withthe plate in quite a small area. The pressure on the blade will be high in this case,but the shearing force needed may not be. This applies to cases where the cuttingangle α is non-zero (1). Considering this, the mechanical cutting of very high-strengthsteels seems to be a potential alternative. It has also been shown, that die cuttingis a possible method for cutting very high strength steels (6).

Figure 1. Schematic figure of the cutting process.

3. Signal processing

The fast Fourier transform (FFT) is very commonly used in order to create a fre-quency spectrum. It is possible, however, to make other use of the FFT. In (7)

Lahdelma defined the real order derivative x(α) of the function x = Xeiωt as follows:

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x(α) = ωαXei(ωt+α π2 ), (1)

where X is a constant, α is the order of derivative, ω is the angular frequency, e isthe Napier’s constant, i is the imaginary unit and t is a real variable representingtime. We can also express (1) in the following form:

x(α) = (iω)αx. (2)

The FFT results in a sequence of complex components {Xk}, which presents ampli-tude and the phase angle of each term of the Fourier series. We can now calculatedthe complex spectrum for the α order derivative as follows:

Xαk = (iωk)αXk (3)

Integration can be understood as ‘a negative derivative’ which can be calculatedusing negative values of α. After performing wanted operations on the sequence,the time domain signal of α order derivative can be obtained by the inverse Fouriertransform (FFT-1).

Filtering can also be done in a slightly similar way. The frequency components,that we want to remove just have to be multiplied by zero. It is important to note,that when the signal is processed in the frequency domain, and resulting the seriesis transformed to time domain by FFT-1, a window function must be used beforeFFT.

4. Weighted lp norms and measurement indices

The weighted lp norm was utilised in order to evaluate the levels of signals and thestress of the cutter. It is defined by:

∥x(α)∥p,w = (N∑

i=1wi|x(α)

i |p)1p , (4)

where the real number α is the order of derivative, x is now displacement, N is thenumber of signal values, and the real number p ̸= 0. If w1 = w2 = ... = wN = 1 thequestion is of a classical lp norm ∥x(α)∥p. Furthermore, if w1 = w2 = ... = wN = 1

N,

we obtain the formula (5). It has the same form as the generalised mean, also knownas the power mean or the Hölder mean. Lahdelma has introduced in (8) the conceptof the space lp. The lp norm is

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∥x(α)∥p ≡ ( 1N

N∑i=1

|x(α)i |p)

1p = ( 1

N)

1p ∥x

(α)i ∥p. (5)

Some results about using this norm for acceleration signals can be found in (8,9).However the signal does not have to be an acceleration signal, since the lp norm canbe calculated regardless of signal type (10,11,12).

It is also possible to use this norm in order to create a dimensionless measurementindex. The MIT measurement index (9) is calculated by

τ MIT p1,p2,...,pnα1,α2,...,αn

= 1n

n∑i=1

bαi

||x(αi)||pi

(||x(αi)||pi)0

(6)

where bαiis the weighting factor selected for order of derivative α, and the norms

||x(αi)||piare calculated from signals x(αi). The norm (||x(αi)||pi

)0 is used as a refer-ence. The reference is often obtained from a machine in good condition.

5. Measurements

The measurements were performed with an accelerometer and a strain gauge. Theaccelerometer was stud-mounted axially to the bearing of the worm in the gear. Thestrain gauge was mounted to the rod of the cutter. A sketch of the cutter with themounting points of sensors is shown in Figure 2.

Figure 2. Schematic figure of the cutter.

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The strain signals are presented as voltage, because we could not reliably calibratethe strain gauge. Therefore we analyzed the relative change in signals and do notwish to evaluate the absolute level of strain.

The signals in Figure 3 are from a situation, where 10 mm thick conventional steelis cut. In The acceleration signal is shown in Figure 3 (a) and the strain signal inFigure 3 (b). Figures 4 and 5 contain corresponding signals, but in Figure 4 theplate cut is 35 mm thick conventional steel, and in Figure 5 the question is of 6 mmthick hardened high-strength steel.

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−20

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Time (s)

x(2) (

m/s

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5Time domain signal

Time (s)

Vol

tage

(V

)

Figure 3. (a) Acceleration signal in the frequency range from 2 to 3000 Hzand (b) strain signal from the frequency range 0 to 40 Hz. Conventionalsteel plate of 10 mm thickness.

(a)

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Time (s)

Vol

tage

(V

)

Figure 4. (a) Acceleration signal in the frequency range from 2 to 3000 Hzand (b) strain signal from the frequency range 0 to 40 Hz. Conventionalsteel plate of 35 mm thickness.

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(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−20

−15

−10

−5

0

5

10

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Time (s)

x(2) (

m/s

2 )

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1

0

1

2

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4

5Time domain signal

Time (s)

Vol

tage

(V

)Figure 5. (a) Acceleration signal in the frequency range from 2 to 3000Hz and (b) strain signal from the frequency range 0 to 40 Hz. Hardenedsteel plate of 6 mm thickness.

The signals in Figures 3 (b), 4 (b) and 5 (b) are proportional to the force needed tocut the plate. To evaluate the relative maximum of the shearing force, we calculatedthe peak-to-peak value from each signal. The peak-to-peak value was used insteadof peak value, because this way we can eliminate the effect of the offset in signals.

In Figures 6, 7 and 8 we can see the same signals after some further signal processing.Both acceleration and strain signals are filtered to contain frequencies between 2 Hzand 40 Hz. In addition, the acceleration signals have been integrated twice in orderto obtain displacement signals.

As one can easily determine, the displacement signals and the strain signals in thefrequency range from 2 to 40 Hz resemble each other quite considerably. Thus itis quite reasonable to conclude that both sensors detect the effect of the cuttingprocess. The strain on the rod of the cutter is relative to shearing force, so one candetermine that the displacement measured on the worm gear is mainly caused bythe very same force.

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(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−800

−600

−400

−200

0

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Time (s)

x (µ

m)

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1

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−0.4

−0.2

0

0.2

0.4

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1Time domain signal

Time (s)

Vol

tage

(V

)Figure 6. (a) Displacement signal and (b) strain signal in the the fre-quency range from 2 to 40 Hz. Conventional steel plate of 10 mm thick-ness.

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−800

−600

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Time (s)

x (µ

m)

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0

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1Time domain signal

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tage

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Figure 7. (a) Displacement signal and (b) strain signal in the the fre-quency range from 2 to 40 Hz. Conventional steel plate of 35 mm thick-ness.

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(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−800

−600

−400

−200

0

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Time (s)

x (µ

m)

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1

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−0.2

0

0.2

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1Time domain signal

Time (s)

Vol

tage

(V

)Figure 8. (a) Displacement signal and (b) strain signal in the the fre-quency range from 2 to 40 Hz. Hardened steel plate of 6 mm thickness.

6. Analysis

In the analysis of the signals, cutting the 10 mm thick conventional steel was usedas a reference. This is because the material properties of the plate are comparableto steels that have been cut by means of the cutter for decades, and the plate isrelatively thin. To determine shearing force relative to this case, we used peak-to-peak values of the strain signal. As it can be seen from the Table 1, the force neededto cut the plate is greatest when cutting 35 mm thick plate. Difference to otherplates is significant.

Table 1. Peak-to-peak values of the strain signals, and their relativechanges compared to cutting 10 mm thick conventional steel.

Plate Peak-to-peak value Relative change10 mm conventional steel 0.922 V 1.00035 mm conventional steel 4.357 V 4.726

6 mm hardened steel 0.417 V 0.452

The yield strength of this hardened steel is around 1000 MPa, where that of con-ventional steel is around 355 MPa. As shown in Table 1, the maximum force whencutting 6 mm thick hardened steel is less than a half of the case where 10 mm conven-tional steel is cut. This may seem odd, but this outcome has a logical explanation,which was presented in Section 2.

For further analysis, the S surface (13) is utilised. The S surface is created by calcu-lating the values of different MIT indices (6) with different values of α and p when

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n = 1 and weighting factor is 1, and plotting the results to a 3D diagram. We haveused a step of 0.25 for α and p, when α ∈ [−4, 4] and p ∈ [0.25, 10].

Figures 9 and 10 show the S surfaces from the comparison of cutting 35 mm and 10mm thick conventional steel. As it was also seen from the peak-to-peak values inTable 1, the maximum tension is almost 5 times higher when cutting 35 mm plate.This can be considered a very clear indication of a change, though it seems thatthe acceleration signal provides an even more sensitive way of detecting stress. Asshown bu Figure 9 (a) ,the MIT index value obtained from the acceleration signalfrom the frequency range 2...40 Hz is around 7 when α is roughly 3 and p is 6 orgreater. Figure 9 (b) shows the highest value of MIT obtained from the strain signalis approximately 5.

(a)

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Order of derivativeOrder of norm

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Figure 9. The S surfaces when comparing the cutting of 35 mm thickconventional steel with the cutting 10 mm thick conventional steel. Thesignals are in the frequency range from 2 to 40 Hz, surface (a) is obtainedfrom the acceleration signal and (b) from the strain signal.

It is shown in Figure 10 (a), that the change of plate thickness from 10 mm to 35mm can be indicated even better on the basis of the acceleration signal at frequencyrange from 2 to 3000 Hz, when α is around 1 and p is about 5 or greater.

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(a)

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Figure 10. The S surfaces when comparing the cutting of 35 mm thickconventional steel with the cutting 10 mm thick conventional steel. Thesignals are in the frequency range from 2 to 3000 Hz, surface (a) is ob-tained from the acceleration signal and (b) from the strain signal.

Figures 11 (b) and 12 (b) show that there are only minor changes at most of theMIT indices when the cutting of 10 mm conventional steel is compared with thecutting of 6 mm thick hardened steel. Strain gauge signals show no major changeand almost all of the values of the MIT indices are less than 1. The values of theMIT indices of the acceleration signal at high values of α are high, even around 15.Because the only MIT indices that show a major change are those of a high orderof derivative, it can be stated that in this case only the high frequency vibrationsare increased.

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(a)

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Figure 11. The S surfaces when comparing the cutting of 10 mm thickconventional steel with the cutting of 6 mm thick hardened steel. Thesignals are in the frequency range from 2 to 40 Hz, surface (a) is obtainedfrom the acceleration signal and (b) from the strain signal.

(a)

−4−2

02

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Figure 12. The S surfaces when comparing the cutting of 10 mm thickconventional steel with the cutting of 6 mm thick hardened steel. The sig-nals are in the frequency range from 2 to 3000 Hz, surface (a) is obtainedfrom the acceleration signal and (b) from the strain signal.

7. Conclusions

It is clearly shown that both an accelerometer and a strain gauge can be used detectcertain changes in operational condition. An accelerometer cannot directly measure

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shearing force, but the acceleration signal can, however, be used in stress evaluation.An accelerometer has a response at higher frequencies than does a strain gauge.This means that an accelerometer can detect shock-like events and high-frequencyvibrations, that may cause fatigue in machinery. As shown in Figure 12, too, astrain gauge used in this case was not capable of measuring vibrations of this kind.

In this case an accelerometer seems to be capable of detecting the change in thestress level of the cutter even better than a strain gauge. On the other hand,an accelerometer is not especially suitable for detecting vibrations of a very lowfrequency unless the sensitivity of the sensor is very high. If the cutting does notcause any shocks or high-frequency vibrations, a strain gauge can still detect tensiondeformation in the structure. In this case, however, an accelerometer seems to giveadequate information on the stress level of the cutter. The stress level can also bequite well evaluated using strain signals, with the exception of detecting vibrationsat high frequency.

It can also be concluded, that cutting high-strength steel plates of at least somethickness is possible with mechanical cutters. However, this result can not be ex-trapolated to other cutters without further investigation, though it has been shownthat the shearing force needed may be very low even if the strength of the materialis high. It is also shown, that cutting hardened steel can cause high frequency vi-brations. These vibrations do not cause high tensions, but repetitive shocks mightcause fatigue failure to machine components such as bearings or gears. It is alsopossbile that the rapid wear of the blades makes the cutting of high-strength steelsunprofitable.

References

1. A I Tselikov and V V Smirnov, ‘Rolling mills’, Pergamon Press, Oxford, UK,First English edition, 1965.

2. D Schmoekel and P Gredy, ‘Verbesserung der Schnittteilequalität durchHochgeschwindigkeits-Scherschneiden HGSS’, EFB-Forschungsbericht Nr. 43,Europäische Forschungsgesellschaft für Blechverarbeitung, Hannover, 1992, [InGerman].

3. R Neugebauer, B Arnold, D Schmoekel and N Geissler, ‘Einfluüsse aufdie Schnittteilqualität beim Scherschneiden von Blechen auf Tafelscheren’,EFB-Forschungsbericht Nr. 130, Europäische Forschungsgesellschaft fürBlechverarbeitung, Hannover, 1999, [In German].

4. SSAB, ‘Plåthandboken - att konstruera och tillverka i höghållfast plåt. EditionVII’, SSAB Tunnplåt AB, Borlänge, Sweden, 1996, [In Swedish].

5. S L Semiatin, editor, ‘ASM handbook’, volume 14B Metalworking: Sheet Form-ing, ASM International, Materials Park, Ohio, USA, 2006.

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6. B Krönauer, M Hirsch, R Golle, H Hoffmann, M Golle and G Jesner, ‘Beschnei-den von pressgehärteten Blechen - Ultrahochfestes 22MnB5-Blech im Scher-schneidprozess’, ZWF Zeitschrift für wirtschaftlichen Fabrikbetrieb, Vol 105,No 1 – 2, pp 26 – 31, 2010, [In German].

7. S Lahdelma, ‘On the derivative of real number order and its application tocondition monitoring’, Kunnossapito, Vol 17, No 8, pp 39–42, 1997.

8. S Lahdelma and J Laurila, ‘Detecting misalignment of a claw clutch using vi-bration measurements’, In Proceedings of CM 2012/MFPT 2012, The NinthInternational Conference on Condition Monitoring and Machinery Failure Pre-vention Technologies, London, UK, 2012.

9. S Lahdelma and E Juuso, ‘Signal processing in vibration analysis’, In Proceed-ings of CM 2008/MFPT 2008, The Fifth International Conference on Condi-tion Monitoring and Machinery Failure Prevention Technologies, pp 879–889,Edinburgh, UK, 2008.

10. S Lahdelma and E Juuso, ‘Signal processing and feature extraction by usingreal order derivatives and generalised norms. Part 1: Methodology’, The Inter-national Journal of Condition Monitoring, Vol 1, No 2, pp 46–53, 2011.

11. S Lahdelma and E Juuso, ‘Signal processing and feature extraction by usingreal order derivatives and generalised norms. Part 2: Applications’, The Inter-national Journal of Condition Monitoring, Vol 1, No 2, pp 54–66, 2011.

12. P S Bullen, ‘Handbook of means and their inequalities’, Kluwer AcademicPublishers, Dordrecht, The Netherlands, 2nd edition, 2003.

13. J Immonen and S Lahdelma, ‘Planeettavaihde - vaativa kunnonvalvontakohde’,Promaint, Vol 27, No 1, pp 34–37, 2013, [In Finnish].

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