Modelling of Oblique Wire Cutting and Experimental ...downloads.hindawi.com/journals/jfq/2019/5429093.pdf · Research Article Modelling of Oblique Wire Cutting and Experimental Application

Research ArticleModelling of Oblique Wire Cutting and ExperimentalApplication on Soft Solid Foods for the Investigation ofFriction Behaviour

Tilman Witt Yannic Hollander Sven Tietze and Jens-Peter Majschak

Chair of Processing Machines and Processing Technology Technische Universitat Dresden 01062 Dresden Germany

Correspondence should be addressed to Tilman Witt tilmanwitttu-dresdende

Received 20 February 2019 Accepted 24 April 2019 Published 16 June 2019

Academic Editor Susana Fiszman

Copyright copy 2019 Tilman Witt et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e sales value of ready-made foods is determined to a large extent by the appearance of the individual pieces or slices Breakoutsor deformations are perceived negatively and should be avoided e quality of the cut surface is derived as an evaluation criterionfrom the evaluation of the cutting force curve is study examines the inuence of friction during cutting emethod developedfor this purpose deals with the problem of separating friction from breaking and deforming when cutting with wire To test thetheoretical approach cutting experiments with thin wires are carried out on various foodstus When cutting with obliquelyoriented wire force value curves are recorded in the feed direction and perpendicular to the feed direction e cutting forcecurves are reduced to two characteristic values for the active unit depending on the cross-section geometry In connection with theresult values an explanatory model is proposed with which a quantied statement on the proportion of friction during the cut ispossible e enhanced model elucidation by oblique cutting will be helpful for the comparison of cutting tools and for theverication of computer simulations

1 Introduction

e cutting process of food is inuenced by productproperties the geometry of the cutting tool and thetechnical parameters of the tool movement e texture offood is characterised by a low Youngrsquos Modulus and oftendistinctive adhesive behaviour [1 2] Foodstus thereforedier signicantly from metallic and polymeric materials e knives for cutting food are of small thickness and asmall wedge angle e oblique cutting movement with ahigh push-slice ratio prevents crushing of the food [3] eappearance and roughness of the cut surface is an im-portant quality criterion for foods [4] which is associatedwith cutting force and sharpness [5 6] e BladeSharpness Index is determined by the tip radius the wedgeangle and the blade prole [7 8] In the case of wirecutting the tip Radii is equivalent to half the wire diameterAfter penetration into the material additional frictionoccurs between the tool and the sample [1] Cutting withwire is often used for products where the friction increases

rapidly as the contact surface increases Despite the smallsurface area of the wire a residual friction remains [9] epresent study deals with the experimental investigation ofwire cutting with oblique wire on foodstus e aim of thepresent study is the quantitative description of the fric-tional force that occurs during a cut

Atkins [10] showed in his investigations on cheesecutting that the interacting surface has a decisive inuenceon the resulting friction force e friction coecient musttherefore be set in relation to the interacting surface and theacting normal force e investigations of Kamyab [9] andGoh [6] deal with the wire cutting of cheese under con-sideration of friction Kamyab determines the coecient offriction by a classical friction test Goh uses a substitute testfrom the investigation of Charalambides [11] In this studyit is assumed that friction can only occur against the di-rection of motion is assumption shall be used as an al-ternative possibility to determine friction during cutting Inthe model of Atkins [10 12] friction is considered in ageneral equation (1) for work during cutting

HindawiJournal of Food QualityVolume 2019 Article ID 5429093 9 pageshttpsdoiorg10115520195429093

W Wfracture + Wplasticity + Wfriction (1)

For food cutting Dowgiallo [13] replaced the term forplastic deformation by elastic and plastic deformationduring cutting By van Vliet [14] further components forviscous cutting have been introduced )e simplifiedequation according to Atkins et al [10] for the pressing cutincludes the separation and displacement of the material inone parameter R and the friction in another term )e depthof the material is considered in equation (2) in parameter w

and the travelled cutting path s )e wire is considered as aline in the model

dW R middot w ds + d(friction) (2)

)ere are several studies that have investigated thefriction that occurs during the cutting of food [1 10 15]Atkins found that by increasing the angle λ the cuttingforce decreases to a local minimum and then increases

again )is property is attributed to the occurrence offriction )e forces in feed direction FMD and perpen-dicular direction FCD are considered for further in-vestigation of friction )e directions of the forces areshown in Figure 1

)ere is no relative movement in the perpendiculardirection which is why instead of mechanical work theforce curves are compared on the basis of the fracturetoughness R Equation (2) is therefore reduced to

dF R dw (3)

)ewidth to be cut depends on the feed distance sfeed ofthe wire By specifying the sample geometry in length wSand height hS the calculation of the cut sample width w canbe calculated as a function of the feed rate and the point offirst contact sI of the cutting tool with the sample A di-vision into three sections is necessary for the cuttingprocess )e determination of the section boundaries isdemonstrated in

w sfeed( 1113857

wS middotsfeed minus sI

tan λ middot wS sI le sfeed le sI + tan λ middot wS

wS sI + tan λ middot wS le sfeed le sI + hS

wS middot 1minussfeed minus sI minus hS

tan λ middot wS1113888 1113889 sI + hS le sfeed le sI + hS + tan λ middot wS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

With w equation (5) for FMD can be formulated

FMD sfeed( 1113857 RMD middot w sfeed( 1113857 (5)

Chaiaia [16] postulates in his investigations a hydrostaticcore along the cutting edge )e proportion of the hydro-static pressure on the edge in perpendicular force-acceptingdirection FCD requires the reference to projected cuttingedge length )e projected cutting edge length is determinedin equation (6) by the tangent of the angle λ

FCD sfeed( 1113857 RCD middot tan λ middot w sfeed( 1113857 (6)

)e angle λ 0deg does not lead to a defined solution for thecalculation of w because of the division by zero)e solutionof the problem by a case distinction is avoided by the in-significant change from λ 0deg to λ 01deg With the equationapproach force curves for FMD and FCD are fitted to themeasured curves Figure 2 qualitatively illustrates the forcecurves for a pressing and an oblique cut

)is model of cutting contains numerous simplifica-tions)e considerations do not take into account the specialfeatures of the initial section and the compression present)e material is regarded as rigid in which the cutting toolimmediately creates a crack on impact For the departing ofthe cutting tool from the material an occurring brittlefracture or a drag of the material into the countersupport isnot mapped by the model

2 Materials and Methods

21 Test Station forWire Cutting )e test setup is based ona universal testing machine type Zwick Z020 with auniaxial feed and is equipped with a force sensor type HP011562 )e sensor has a measuring range of plusmn500N anddetects the force in machine feed direction FMD TwoYCZ-133 bending beams with a measuring range of plusmn50 Nare used to measure the force orthogonally crossing thefeed direction )e measured force in the cross directionFCD is synchronized with the force FMD via a light barrier)e arrangement of the sensors within the module in-stalled in the universal testing machine is shown inFigure 3

)e wire can be tensioned into the frame at three dif-ferent angles λ For an angle λ 0deg the wire is orthogonal tothe feed direction Figure 1 shows in position (d) an angleλ 45deg with the solid line and in positions (e) and (f) theangles of 30deg and 0deg as an interrupted line )e specimen lieson the countersupport which has a gap width of 5mmDuring the test the countersupport stands still and the wirewith the sensors is moved at 1 or 2mms feed rate All testsare performed in an air-conditioned environment at anambient temperature of 296K and 50 humidity)e videosV1-V6 in the supplementary material are attached to vi-sualize the experiment

2 Journal of Food Quality

3 Materials

)e experiments are carried out with sausage LeberkaseEdam cheese and bubble gum )e selection of materials ischaracterised by similar Youngrsquos modulus [2 17] and in-dividually distinctive properties Sausage has a high fatcontent and fibres that are longer than the diameter of thecutting tool Leberkase is finer chopped than sausage and hasonly occasionally long fibres Both are characterised by a lowYoungrsquos modulus compared to metal and have been used insimilar investigations [18] Edam cheese is tested because ofits relatively small holes compared to other cheeses and its

strong tendency to stick to blades [19] Bubble gum also has astrong tendency to adhere [20] is ductile and shows a morepronounced viscosity than the other test materials [21] Asthe mechanical properties of bubble gum are verytemperature-dependent [21] two temperatures will be in-vestigated Sausage Leberkase and cheese are precondi-tioned to 2812 K while bubble gum is heated to 3082 K and3132 K respectively Due to the two temperatures for bubblegum there are a total of five different specimens with bubblegum at 3082K hereinafter referred to as ldquobubble gum 1rdquo andbubble gum at 3132 K hereinafter referred to as ldquobubble gum2rdquo All samples are wrapped and stored for at least four hoursat the specified temperature in a cooled incubator )eindividual samples shall have a length of at least 30mm)ecut surface has a width wS of 10mm and different heightsdepending on the material For Leberkase the samples are13mm high whereas all other materials have a height hS of16mm

31Wires forCutting Two different materials each with twodifferent diameters are used for the wires in the experi-ments For the diameters 006mm and 010mm molyb-denum and dimensions 020mm and 040mm Kanthal areused Molybdenum has a Youngrsquos modulus of elasticity of

a

b

c

d

e

f

g

h

s feed

(t)

Figure 3 Design of the experimental unit (a) Load cell HP 011562(b) Bending beams YCZ-133 (c) Frame (d) Wire fastening po-sition λ 45deg (e) Wire fastening position λ 30deg (f ) Wire fasteningposition λ 0deg (g) Specimen (h) Countersupport of the sample

a

s feed

(t)

wS

w (sfeed) λ

FCD

FMD

h S

b

c

d

e

Figure 1 Schematic sketch of the cutting test (a) Uniaxialguidance (b) Framework for tensioning the wire (c) Wire(d) Wire clamping (e) Specimen Feed rate as a function of timesfeed(t) Height hS and width wS of the sample Cutting width w as afunction of the feed sfeed Oblique angle λ )e force in feed di-rection FMD and the force in orthogonal cross direction FCD

Travel feed sfeed in mm

Forc

e FM

D F

CD in

N

c

a

b

Figure 2 Sketched plot of the calculated force as a function of thefeed distance (a) Describes a typical curve for FMD with λ 0deg(b) Describes a typical curve for FMD with λ 45deg (c) Describes atypical curve for FCD with λ 45deg

Journal of Food Quality 3

310GPa [22] while Kanthal has a Youngrsquos modulus of220GPa [23] Kanthal consists of gt69 Fe 205ndash235 CrAl 58 and lt008 C Due to the lower Youngrsquos modulusof Kanthal a comparable elongation is achieved for thethinner wires as for the thicker wires )e wire tension iscalculated and adjusted as a function of the first harmoniceigenfrequency )e adjustment and gradual checking of theeigenfrequency is indirectly ensured by striking the stringand evaluating the acoustics emitted In screening tests witha wire diameter of 080mm strong sample deformations andbreakouts from the samples occurred )e experiments aretherefore limited to wires with a maximum diameter of04mm )e choice of wires ensures comparability with thework of Goh [6] and Kamyab [9]

32 Checking andCalibrating the Force Sensors A significantmeasurement uncertainty can be caused by the additionalload of a sensor due to lateral force bending or torsion Inthe test setup described transverse forces and bendingmoments occur due to a lever and force components that actperpendicular to the direction of measurement )e accu-racy of the sensors is therefore checked by a referencemeasurement with defined forces and defined force di-rection A wire is attached to the middle cutting point of thecutting wire which leads over a deflection pulley and has aweight of known mass attached to the other end)e cuttingwire is moved with the sensors where the direction of pull ofthe attached mass changes )e pulley is considered fric-tionless and the wire massless )e test is carried out at lowspeed to avoid inertial effects Figure 4 shows a sketch of thetest setup

)e parameter loffset is set to 200mm )e stroke sfeed islimited to 600mm For the different weights of 20 g 50 g100 g 200 g and 500 g the test is performed five times eachand the percentage error between expected and measuredvalue is calculated Figure 5 shows an example of a com-parison between the calculated and measured curves of forceFMD and force FCD for a weight of 100 g)e result of the testis plotted on the basis of a weight of 100 g )e error iscalculated as a percentage of the expected and measuredvalue

)e measured force in the feed direction shows a sig-nificant difference to the calculated curve for small angles)e relative error between measured value and real value isless than 4 for forces greater than 01N

33 Consideration of Wire Deflection )e deflection of thewire by the effective force during cutting cannot be avoidedin the test setup described A reduction of the deflection isachieved by increasing the preload of the wire although thepermissible force during cutting is reduced by the fasteroccurrence of plastic deformation of the wire To determinethe deflection of the wire as a function of the measured forceperpendicular to the wire axis the specimen is replaced by aneedle bearing )e wire will deflect on contact with thebearing ring and generate a force on the sensors as shown inFigure 6

For each combination of wire diameter and obliqueangle five repetitions of the test are performed and a cor-rection curve is derived Using the curve the true position of

s feed(t)

a

b

c

de

f

loffset

g

α

Figure 4 Schematic sketch of the test of the force sensors(a) Uniaxial guidance (b) Framework for tensioning the wire(c) )ick wire (d) Clamping wire (e) )inner wire (f ) Pulley(g) Test mass Angle of the acting force in relation to the feed di-rection α Feed as a function of time sfeed(t) Distance between pulleypivot point and feed axis loffset Test mass m attached to a wire

00

ndash02

ndash04

ndash06

Angle α in rad

ndash10

ndash08

18π 14π 38π 12π

Forc

e FM

D F

CD in

N

Figure 5 Plot of the measured forces with respect to the feeddistance over 600mm path at 100 g weight In black is FMD and ingrey FCD )e solid lines represent the measured forces and thedotted lines represent the calculated forces


the wire in the product is calculated back to the pathidentification by the feed axis Figure 7 shows the correctionfor a selected example

)e corrected force curves occur at a wire deflection of0mm A residual error in the correction remains due to theaveraging of five repetitions and leads to minor deviations)emaximum deflection is determined at a wire diameter of006mm and an angle λ 45deg )e deflection of the wire isdetermined by the force vector in perpendicular direction tothe wire axis A map for the compensation of the deflectiondepending on the perpendicular force vector to the wire isestablished after this experiment )e data are used tocorrect the measured cutting force curves in order to obtaina force curve that is as representative as possible

4 Results

41 Comparison of Model and Cutting Force Curve )eperformance of the tests with sausage shows very largedifferences in the cutting forces between the individualsamples Sausage is therefore not considered in the furtherevaluation )e remaining test specimens show comparablebehaviour at similar cutting forces )e comparison of themeasured force curve with the fitted force curve is shownbelow )e point of initial contact between wire and spec-imen is determined according to the criterion of a forcethreshold By the known height and width of the specimenand the angle λ the interacting length w is determined foreach waypoint according to equation (4) )e force curvesshould be approximated by equations (5) and (6) A fitaccording to the criterion of the smallest error squares is

used to obtain the characteristic parameters RMD and RCDFigure 8 shows the force curves of the four remainingmaterials next to each other with the same settings

)e results of the fitting according to Figure 8 are forRMD and RCD values of 77 Jm2 and 56 Jm2 for cheese114 Jm2 and 79 Jm2 for Leberkase 133 Jm2 and 55 Jm2

for bubble gum 1 and 84 Jm2 and 39 Jm2 for bubble gum2)e values for all tests are listed in the supplementary file)e differences between the measured and calculated curveare smaller for cheese in Figure 8 than for the other ma-terials Leberkase follows the principle course of the curvewith force jumps being visible In the observation of theexperiments an irregular speed of the wire can be seenwhile passing through which can also be seen in the videoShortly before the end of the cut the remaining materialruptures which can be seen in the force curve due to anabrupt drop in force )e bubble gums show larger de-viations from the calculated curve when the wire exitsFrom the observations of the slip a stronger pulling of thematerial into the countersupport is recognizable )estatistical comparison of the model and the measurement isillustrated by the standard error of the estimation SEstFigure 9 shows the error values for the feed direction inblack and the error values for the perpendicular crossdirection in grey )e different angles are distinguished bydifferent markers

)e standard error of estimate for Edam cheese Leb-erkase and bubble gum 1 is significantly greater for λ 0degthan for λ 30deg and 45deg with the same wire diameter Forbubble gum 2 this difference cannot be shown For therespective materials and diameters for λ 30deg or 45deg theerrors are on a similar level Bubble gum 1 has an overallhigher average of defects Edam cheese has increasingerrors with increasing wire diameter while Leberkaseshows this tendency only slightly for the transverse forceAs the wire diameter increases the force error in bubble

5

5

4

4

3

3

2

2

0

Wire deflection ∆sfeed in mm0

1

1

Forc

e FM

D F

CD in

N

Figure 7 Plot of forces with respect to the feed difference withλ 30deg )e black continuous and dotted lines represent FMD formeasured and corrected trace )e grey solid and dotted linesrepresent FCD for the measured and corrected trace

a

∆sfeed

s feed(t)

b

c

d

e

f

Figure 6 Schematic sketch of the wire deflection test (a) Uniaxialguidance (b) Framework for tensioning the wire (c) Wiredeflected (d) Wire clamping (e) Wire not deflected (f ) PulleyFeed travel sfeed(t) Wire deflection Δsfeed


gum increases in the feed direction while the force error inthe transverse direction decreases An increase of thestandard error by increasing the feed rate can be observedespecially with bubble gum Edam cheese and Leberkase onthe other hand show only slight increases )e low-temperature bubble gum 1 has a larger mean standarderror than the high-temperature bubble gum 2 )estandard deviation of the standard error of estimate in-creases with increasing standard error For Leberkase thestandard deviation relative to the standard error is higherthan for the other materials

42 Fracture Toughness in Feed Direction and PerpendicularDirection )e fitted values for the factors RMD and RCD areshown in Figure 10 as a function of the feed rate vspeed theoblique angle λ the wire diameter dWire and the material Ameaningful evaluation of RCD is not possible for press cuts)erefore only RMD is displayed for λ 0deg

)e characteristic factors RMD and RCD for the cuttingforce curves are analysed with a regression for the influ-encing variables A mixed model that considers the materialas a random intercept is discarded due to poor elucidationAccordingly a single model is generated for each material

Forc

e FM

D F

CD in

N15


10

05

00

0 10 20 30

(a)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(b)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(c)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(d)

Figure 8 Plot the force in respect to the feed during a cut )e angle λ 45deg and the wire has a diameter of dWire 01mm )e feed ratevspeed 1mms )e FMD is shown in black and the FCD in grey )e dash-dot lines represent the fitted force curves and the continuous linesresult from the corrected measurements (a) Edam Cheese (b) Leberkase (c) Bubble gum 1 (d) Bubble gum 2

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020

000006 010 020 040Wire diameter dWire

in mm by groups

(a)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(b)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(c)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(d)

FIGURE 9 Plot of the standard error of estimation SEst in respect to the wire diameter dWire)e error for FMD is shown in black and for FCD ingrey )e circles represent λ 0deg rectangles for λ 30deg and rhombus for λ 45deg )e error bars represent the standard deviation ofmeasurements (a) Edam Cheese (b) Leberkase (c) Bubble gum 1 (d) Bubble gum 2


Equation (7) shows the model for the value of RMD createdfor each material

RMDMaterial λ vSpeed dWire1113872 1113873

Intercept +βλrad

middot λ +βv middot s

mmmiddot vSpeed +

βdmm

middot dWire1113888 1113889Jm2

(7)

)e regression coefficients and their significance arelisted in Table 1 )e significance is given in steps where thep value must be less than 0001 for three stars and less than001 for two stars For the regression the influences arescaled to the range from 0 to 1 in order to be able to comparethe effect strength

All oblique angles are taken into account when de-termining RMD whereas for RCD the pressing cut is nottaken into account )e recorded influencing variablesare significant for RMD in almost all cases )e influenceof speed in Leberkase is not significant for any of thefactors )e influence of the oblique angles is not sig-nificant for RCD in several cases )e quantitative valuesfor the coefficients reflect the expected increasesaccording to Figure 10 )e scaled coefficients of theoblique angles are significantly lower compared to theother coefficients

5 Discussion

)e increasing standard errors of estimate with increasingwire diameter are due to a stronger deformation of thecutting quality at the entry and depart of the wire )is

applies in particular to the pressing cut in which the wirerests on the entire line and the material is strongly deformedbefore the cut is initiated Bubble gum shows the largestoverall deviations Experiments have shown that chewinggum exhibits pronounced ductile behaviour compared toother materials Both the ductile and the elastic behaviourare not represented by the regression and therefore lead tohigher standard errors

Other studies have shown that Edam cheese andbubble gum have high friction whereas Leberkase hasrelatively low friction )e comparison of RMD with RCDfor Leberkase shows an approximate equality of bothfactors Edam cheese on the other hand shows a colinearincrease in RCD as RMD increases Colinearity cannot beassumed for bubble gum which also has high friction ForEdam cheese and bubble gum there is a clear differencebetween RMD and RCD It is therefore assumed that thisdifference can be explained by the occurrence of frictionFigure 11 sketches the force ratios for a cut with an obliquewire

It is assumed that a surface load can be indicated onthe surface of the round cutting tool as a function of theangle q(φ) Without loss of generality the representationis selected for a specific φ0 )e value for q(φ0) is expressedas normal force dFN(φ0) for illustration purposes )eparticles of the test material move in a plane that can bedetermined by the coordinates of the contact point be-tween the cutting tool and the test material and thetangential plane at the contact point on the surface of thecutting tool In Figure 11 the plane is given by the vectorscWire and tWire and the contact point Perpendicular to theplane in the contact point the force dFN(φ0) is shown )e

300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)

Figure 10 Plot of the factors of the regression RMD RCD in relation to the wire diameters dWire )e values for RMD are shown in black andfor RCD in grey)e circles represent λ 0deg rectangles for λ 30deg and rhombus for λ 45deg)e error bars represent the standard deviation ofmeasurements (a) Edam Cheese (b) Leberkase (c) Bubble gum 1 (d) Bubble gum 2


direction of the feed sfeed forces the particle to move in thetangential plane Since there are other particles above andbelow the particle the movement on the surface will benormal to the cutting surface )e normal vector of thecutting surface ex the vector in feed direction sfeed and thecontact point define the plane of motion )e intersection ofthe tangential plane and the motion plane of the particleresults in the motion line of the particle )e line of motion isparallel to the displayed tangent tWire )e force vectordFR(φ0) is located on this line )e splitting of the forcevectors along the Cartesian coordinate system allows thedirectional summation of the force components )e com-ponents in x direction cancel each other out due to symmetry)e part in z direction is generated exclusively by the normalforce whereas the force in y direction is influenced by normalforce and friction force Formulas (8)ndash(10) summarize thedependencies for the x y and z directions

Fx FNx + FRx 1113946π

0q(φ) middot sinφ dφ

+ 1113946π

0q(φ) middot μ middot sin arctan

tanφcos λ

1113874 11138751113874 1113875dφ

(8)

Fy FNy + FRy cos λ middot 1113946π

0q(φ) middot cosφ dφ

+ 1113946π

0q(φ) middot μ middot cos arctan

tanφcos λ

1113874 11138751113874 1113875dφ

(9)

Fz FNz sin λ middot 1113946π

0q(φ) middot cosφ dφ (10)

An evaluation in x direction is not possible due to thesymmetry Since q(φ) and friction coefficient μ are unknownthe integral parts in equations (11) and (12) are substitutedby Ffric and Ffracdisp

Fy cos λ middot Ffracdisp + Ffric (11)

Fz sin λ middot Ffracdisp (12)

)e friction force Ffric and the force corresponding tofraction and displacement Ffracdisp can be determined if Fyand Fz are known )e force values were recorded in theexperiments An equation of the force in motion righting bythe equation of the experiment 5 and equation of the modelapproach 12 can be converted to equation (13) )e sameapplies to perpendicular to the direction of motion byequating equations (6) and (12) to equation (14)

Ffric RMD middot w sfeed( 1113857minus cos λ middot Ffracdisp (13)

Ffracdisp RCD middot1

cos λmiddot w sfeed( 1113857 (14)

)e methodical procedure enables the comparisonbetween active units with regard to their friction behaviour)e quantitative evaluation during the cut has the ad-vantage that the friction can also be determined for normalforces at which the material to be cut is destroyed )emethod thus represents a possibility for determining thefriction Computer simulations are recommended forfurther investigation and confirmation of the methodo-logical approach

SymmetryPlane y times z

dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)

dq(φ0)dφ = dFN(φ0)tWire

cWire

sfeed

sfeed

Figure 11 Half symmetrical sketch of the acting forces and theirCartesian components on the surface of an oblique wire )eangle of the slope is indicated by λ )e feed direction is rep-resented by sfeed )e line load q(φ0) describes the load along thewire circumference For a specific φ0 the normal force dFN(φ0) atthe incremental contact point is represented in perpendiculardirection to the wire surface )e tangential wire surface is de-scribed by the tangent tWire and the rotation axis of the wire cWire)e friction force dFR acts in the direction of tWire )e forcesdFN(φ0) and dFR(φ0) are also given in their Cartesiancomponents

Table 1 Model influences with significance rating

Coefficient Edam cheese Leberkase Bubble gum 1 Bubble gum 2

RMD in Jm2

Intercept minus1279 lowastlowast 7124 lowastlowastlowast 2570 lowastlowastlowast 1780 lowastlowastlowast

βλ 1221 lowastlowastlowast 1447 lowastlowastlowast 1799 lowastlowastlowast 1097 lowastlowastlowast

βv 2533 lowastlowastlowast minus104 6560 lowastlowastlowast 4205 lowastlowastlowast

βd 46433 lowastlowastlowast 5422 lowastlowastlowast 25795 lowastlowastlowast 13184 lowastlowastlowast

RCD in Jm2

Intercept minus947 5102 lowastlowastlowast 3972 lowastlowastlowast 1296 lowastlowastlowast

βλ 458 minus262 minus1889 lowastlowastlowast 283βv 1977 lowastlowastlowast minus047 2760 lowastlowastlowast 1909 lowastlowastlowast


lowastlowastlowastplt 0001 lowastlowastplt 001


6 Conclusions

)e additional recording of the force curve in orthogonaldirection to the cutting direction enables the evaluation of asecond equation and a higher clarification in the modelapproach for the description of cutting )e implementationand evaluation of a test series provides quantitative valuesfor testing the discussed model approach )e limits of themodel are investigated in experiments whereby especiallymaterials with high tensile strength and low complexmodulus are insufficiently represented )e proportion offriction during cutting can be determined by evaluating theforce curve in the feed direction and the force curve in theperpendicular direction )e methodical approach makes itpossible to identify measures for reducing the cutting forcesassociated with an increase in cutting quality

Data Availability

)e recorded characteristic values used to support thefindings of this study are included within the supplementaryinformation files

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by funds from the European SocialFund and cofinanced by tax revenues on the basis of thebudget approved by the members of the Saxon State Par-liament and by funds from)eegarten-Pactec GmbH amp CoKG We acknowledge the support by the Open AccessPublication Funds of the SLUBTU Dresden

Supplementary Materials

)e characteristic test values per section are provided as a listin a CSV file Six videos of the cuts are included to visualizethe experiments (Supplementary Materials)

References

[1] Y Schneider S Zahn C Schindler and H Rohm ldquoUltrasonicexcitation affects friction interactions between food materialsand cutting toolsrdquo Ultrasonics vol 49 no 6-7 pp 588ndash5932009

[2] C Schmidt R Bornmann S Schuldt Y Schneider andH Rohm ldquo)ermo-mechanical properties of soft candyapplication of time-temperature superposition to mimic re-sponse at high deformation ratesrdquo Food Biophysics vol 13no 1 pp 11ndash17 2018

[3] T Atkins ldquoOptimum blade configurations for the cutting ofsoft solidsrdquo Engineering Fracture Mechanics vol 73 no 16pp 2523ndash2531 2006

[4] Y Schneider S Zahn and L Linke ldquoQualitative processevaluation for ultrasonic cutting of foodrdquo Engineering in LifeSciences vol 2 no 6 pp 153ndash157 2002

[5] A G Atkins and J F V Vincent ldquoAn instrumented mi-crotome for improved histological sections and the mea-surement of fracture toughnessrdquo Journal of Materials ScienceLetters vol 3 no 4 pp 310ndash312 1984

[6] S M Goh M N Charalambides and J G Williams ldquoOn themechanics of wire cutting of cheeserdquo Engineering FractureMechanics vol 72 no 6 pp 931ndash946 2005

[7] G A Reilly B A O McCormack and D Taylor ldquoCuttingsharpness measurement a critical reviewrdquo Journal of Mate-rials Processing Technology vol 153-154 pp 261ndash267 2004

[8] C T McCarthy A N Annaidh and M D Gilchrist ldquoOn thesharpness of straight edge blades in cutting soft solids partIImdashanalysis of blade geometryrdquo Engineering Fracture Me-chanics vol 77 no 3 pp 437ndash451 2010

[9] I Kamyab S Chakrabarti and J GWilliams ldquoCutting cheesewith wirerdquo Journal of Materials Science vol 33 no 11pp 2763ndash2770 1998

[10] A G Atkins X Xu and G Jeronimidis ldquoCutting by lsquopressingand slicingrsquo of thin floppy slices of materials illustrated byexperiments on cheddar cheese and salamirdquo Journal of Ma-terials Science vol 39 no 8 pp 2761ndash2766 2004

[11] M N Charalambides S M Goh S L Lim and J GWilliamsldquo)e analysis of the frictional effect on stressmdashstrain datafrom uniaxial compression of cheeserdquo Journal of MaterialsScience vol 36 no 9 pp 2313ndash2321 2001

[12] A G Atkins ldquoFracture toughness and cuttingrdquo InternationalJournal of Production Research vol 12 no 2 pp 263ndash2741974

[13] A Dowgiallo ldquoCutting force of fibrous materialsrdquo Journal ofFood engineering vol 66 no 1 pp 57ndash61 2005

[14] T van Vliet ldquoLarge deformation and fracture behaviour ofgelsrdquo Current Opinion in Colloid amp Interface Science vol 1no 6 pp 740ndash745 1996

[15] T Brown S J James and G L Purnell ldquoCutting forces infoods experimental measurementsrdquo Journal of food engi-neering vol 70 no 2 pp 165ndash170 2005

[16] B Chiaia ldquoFracture mechanisms induced in a brittle materialby a hard cutting indenterrdquo International Journal of Solids andstructures vol 38 no 44-45 pp 7747ndash7768 2001

[17] K R Agrawal P W Lucas J F Prinz and I C BruceldquoMechanical properties of foods responsible for resisting foodbreakdown in the human mouthrdquo Archives of Oral Biologyvol 42 no 1 pp 1ndash9 1997

[18] S Schuldt Y Schneider andH Rohm ldquoHigh-speed cutting offoods cutting behavior and initial cutting forcesrdquo Journal ofFood Engineering vol 230 pp 55ndash62 2018

[19] G Arnold L Leiteritz S Zahn and H Rohm ldquoUltrasoniccutting of cheese composition affects cutting work reductionand energy demandrdquo International Dairy Journal vol 19no 5 pp 314ndash320 2009

[20] K Nevzat I Palabiyik S O Toker and O Sagdic ldquoChewinggum production quality parameters and opportunities fordelivering bioactive compoundsrdquo Trends in Food Science ampTechnology vol 55 pp 29ndash38 2016

[21] S Schuldt T Witt C Schmidt et al ldquoHigh-speed cutting offoods development of a special testing devicerdquo Journal ofFood Engineering vol 216 pp 36ndash41 2018

[22] R Farraro and R B Mclellan ldquoTemperature dependence ofthe Youngrsquos modulus and shear modulus of pure nickelplatinum and molybdenumrdquo Metallurgical Transactions Avol 8 no 10 pp 1563ndash1565 1977

[23] A Saigal and G G Leisk ldquoResidual strains and stresses intungstenkanthal compositesrdquo Materials Science and Engi-neering A vol 237 no 1 pp 65ndash71 1997


Hindawiwwwhindawicom

International Journal of

Volume 2018

Zoology

Hindawiwwwhindawicom Volume 2018

Anatomy Research International

PeptidesInternational Journal of



Journal of Parasitology Research

GenomicsInternational Journal of


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018


BioinformaticsAdvances in

Marine BiologyJournal of



Neuroscience Journal


BioMed Research International

Cell BiologyInternational Journal of



Biochemistry Research International

ArchaeaHindawiwwwhindawicom Volume 2018


Genetics Research International


Advances in

Virolog y Stem Cells International



Enzyme Research



MicrobiologyHindawiwwwhindawicom

Nucleic AcidsJournal of

Volume 2018

Submit your manuscripts atwwwhindawicom

W Wfracture + Wplasticity + Wfriction (1)

For food cutting Dowgiallo [13] replaced the term forplastic deformation by elastic and plastic deformationduring cutting By van Vliet [14] further components forviscous cutting have been introduced )e simplifiedequation according to Atkins et al [10] for the pressing cutincludes the separation and displacement of the material inone parameter R and the friction in another term )e depthof the material is considered in equation (2) in parameter w

and the travelled cutting path s )e wire is considered as aline in the model

dW R middot w ds + d(friction) (2)

)ere are several studies that have investigated thefriction that occurs during the cutting of food [1 10 15]Atkins found that by increasing the angle λ the cuttingforce decreases to a local minimum and then increases

again )is property is attributed to the occurrence offriction )e forces in feed direction FMD and perpen-dicular direction FCD are considered for further in-vestigation of friction )e directions of the forces areshown in Figure 1

)ere is no relative movement in the perpendiculardirection which is why instead of mechanical work theforce curves are compared on the basis of the fracturetoughness R Equation (2) is therefore reduced to

dF R dw (3)

)ewidth to be cut depends on the feed distance sfeed ofthe wire By specifying the sample geometry in length wSand height hS the calculation of the cut sample width w canbe calculated as a function of the feed rate and the point offirst contact sI of the cutting tool with the sample A di-vision into three sections is necessary for the cuttingprocess )e determination of the section boundaries isdemonstrated in

w sfeed( 1113857

wS middotsfeed minus sI

tan λ middot wS sI le sfeed le sI + tan λ middot wS

wS sI + tan λ middot wS le sfeed le sI + hS

wS middot 1minussfeed minus sI minus hS

tan λ middot wS1113888 1113889 sI + hS le sfeed le sI + hS + tan λ middot wS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

With w equation (5) for FMD can be formulated

FMD sfeed( 1113857 RMD middot w sfeed( 1113857 (5)

Chaiaia [16] postulates in his investigations a hydrostaticcore along the cutting edge )e proportion of the hydro-static pressure on the edge in perpendicular force-acceptingdirection FCD requires the reference to projected cuttingedge length )e projected cutting edge length is determinedin equation (6) by the tangent of the angle λ

FCD sfeed( 1113857 RCD middot tan λ middot w sfeed( 1113857 (6)

)e angle λ 0deg does not lead to a defined solution for thecalculation of w because of the division by zero)e solutionof the problem by a case distinction is avoided by the in-significant change from λ 0deg to λ 01deg With the equationapproach force curves for FMD and FCD are fitted to themeasured curves Figure 2 qualitatively illustrates the forcecurves for a pressing and an oblique cut

)is model of cutting contains numerous simplifica-tions)e considerations do not take into account the specialfeatures of the initial section and the compression present)e material is regarded as rigid in which the cutting toolimmediately creates a crack on impact For the departing ofthe cutting tool from the material an occurring brittlefracture or a drag of the material into the countersupport isnot mapped by the model

2 Materials and Methods

21 Test Station forWire Cutting )e test setup is based ona universal testing machine type Zwick Z020 with auniaxial feed and is equipped with a force sensor type HP011562 )e sensor has a measuring range of plusmn500N anddetects the force in machine feed direction FMD TwoYCZ-133 bending beams with a measuring range of plusmn50 Nare used to measure the force orthogonally crossing thefeed direction )e measured force in the cross directionFCD is synchronized with the force FMD via a light barrier)e arrangement of the sensors within the module in-stalled in the universal testing machine is shown inFigure 3

)e wire can be tensioned into the frame at three dif-ferent angles λ For an angle λ 0deg the wire is orthogonal tothe feed direction Figure 1 shows in position (d) an angleλ 45deg with the solid line and in positions (e) and (f) theangles of 30deg and 0deg as an interrupted line )e specimen lieson the countersupport which has a gap width of 5mmDuring the test the countersupport stands still and the wirewith the sensors is moved at 1 or 2mms feed rate All testsare performed in an air-conditioned environment at anambient temperature of 296K and 50 humidity)e videosV1-V6 in the supplementary material are attached to vi-sualize the experiment


3 Materials




a

b

c

d

e

f

g

h

s feed

(t)


a

s feed

(t)

wS

w (sfeed) λ

FCD

FMD

h S

b

c

d

e



Forc

e FM

D F

CD in

N

c

a

b









s feed(t)

a

b

c

de

f

loffset

g

α


00

ndash02

ndash04

ndash06

Angle α in rad

ndash10

ndash08

18π 14π 38π 12π

Forc

e FM

D F

CD in

N





4 Results






5

5

4

4

3

3

2

2

0


1

1

Forc

e FM

D F

CD in

N


a

∆sfeed

s feed(t)

b

c

d

e

f






Forc

e FM

D F

CD in

N15


10

05

00

0 10 20 30

(a)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(b)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(c)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(d)


060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(a)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(b)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(c)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(d)





Intercept +βλrad


mmmiddot vSpeed +

βdmm


(7)



5 Discussion





300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)






+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018


3 Materials




a

b

c

d

e

f

g

h

s feed

(t)


a

s feed

(t)

wS

w (sfeed) λ

FCD

FMD

h S

b

c

d

e



Forc

e FM

D F

CD in

N

c

a

b









s feed(t)

a

b

c

de

f

loffset

g

α


00

ndash02

ndash04

ndash06

Angle α in rad

ndash10

ndash08

18π 14π 38π 12π

Forc

e FM

D F

CD in

N





4 Results






5

5

4

4

3

3

2

2

0


1

1

Forc

e FM

D F

CD in

N


a

∆sfeed

s feed(t)

b

c

d

e

f






Forc

e FM

D F

CD in

N15


10

05

00

0 10 20 30

(a)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(b)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(c)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(d)


060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(a)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(b)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(c)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(d)





Intercept +βλrad


mmmiddot vSpeed +

βdmm


(7)



5 Discussion





300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)






+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018








s feed(t)

a

b

c

de

f

loffset

g

α


00

ndash02

ndash04

ndash06

Angle α in rad

ndash10

ndash08

18π 14π 38π 12π

Forc

e FM

D F

CD in

N





4 Results






5

5

4

4

3

3

2

2

0


1

1

Forc

e FM

D F

CD in

N


a

∆sfeed

s feed(t)

b

c

d

e

f






Forc

e FM

D F

CD in

N15


10

05

00

0 10 20 30

(a)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(b)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(c)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(d)


060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(a)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(b)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(c)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(d)





Intercept +βλrad


mmmiddot vSpeed +

βdmm


(7)



5 Discussion





300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)






+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018




4 Results






5

5

4

4

3

3

2

2

0


1

1

Forc

e FM

D F

CD in

N


a

∆sfeed

s feed(t)

b

c

d

e

f






Forc

e FM

D F

CD in

N15


10

05

00

0 10 20 30

(a)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(b)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(c)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(d)


060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(a)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(b)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(c)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(d)





Intercept +βλrad


mmmiddot vSpeed +

βdmm


(7)



5 Discussion





300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)






+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018





Forc

e FM

D F

CD in

N15


10

05

00

0 10 20 30

(a)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(b)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(c)


Forc

e FM

D F

CD in

N

15

10

05

00

0 10 20 30

(d)


060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(a)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(b)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(c)

060

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Stan

dard

erro

r of e

stim

atio

n S E

st in

N 040

020

000

060

040

020


in mm by groups

(d)





Intercept +βλrad


mmmiddot vSpeed +

βdmm


(7)



5 Discussion





300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)






+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018




Intercept +βλrad


mmmiddot vSpeed +

βdmm


(7)



5 Discussion





300v S

peed

= 2

mm

sv S

peed

= 1

mm

sFa

ctor

s RM

D R

CD in

Jm

2


in mm by groups

200

100

0

300

200

100

0

(a)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(b)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(c)


in mm by groups

300

v Spe

ed =

2m

ms

v Spe

ed =

1m

ms

Fact

ors R

MD

RCD

in J

m2

200

100

0

300

200

100

0

(d)






+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018





+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(8)



+ 1113946π


tanφcos λ

1113874 11138751113874 1113875dφ

(9)












dFRy(φ0)

dFRx(φ0)

dFR(φ0)z

x y

dFNy(φ0)

q(φ0)

φ0

dFNx(φ0)

dFNz(φ0)


cWire

sfeed

sfeed




RMD in Jm2





RCD in Jm2






6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018


6 Conclusions


Data Availability




Acknowledgments




References



























Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018




Volume 2018

Zoology











Volume 2018

















Advances in




Enzyme Research





Volume 2018


Documents

Modelling of Oblique Wire Cutting and Experimental ...downloads.hindawi.com/journals/jfq/2019/5429093.pdf · Research Article Modelling of Oblique Wire Cutting and Experimental Application