Dpt. Of Mechanical Engineering Primer Congreso ...ehu.eus/manufacturing/docpubli/200.pdf · Dpt. Of...

Dpt. Of Mechanical Engineering

UNIVERSIDAD DEL PAIS VASCO

CISIFCISIF--MESICMESIC

20052005

Optimización del Fresado de Piezas de Baja Optimización del Fresado de Piezas de Baja Rigidez mediante un Modelo de Estabilidad Rigidez mediante un Modelo de Estabilidad

TridimensionalTridimensional

Optimización del Fresado de Piezas de Baja Rigidez Optimización del Fresado de Piezas de Baja Rigidez mediante un Modelo de Estabilidad Tridimensionalmediante un Modelo de Estabilidad Tridimensional

L.N. López de Lacalle*, L.N. López de Lacalle*, F. J. CampaF. J. Campa, S. Herranz , U. Bravo, S. Herranz , U. Bravo

*. Department of Mechanical Engineering of the University of the*. Department of Mechanical Engineering of the University of the Basque Country.Basque Country.

+. FATRONIK.FATRONIK.

Primer Congreso Internacional Primer Congreso Internacional

de la Sociedad de Ingeniería de Fabricaciónde la Sociedad de Ingeniería de Fabricación

Calatayud Septiembre2005




20052005



IntroductionIntroduction

ThreeThree dimensional dimensional stabilitystability modelmodel

Experimental Experimental ValidationValidation

ConclusionsConclusions

FutureFuture worksworks

Contents




20052005



Introduction

MONOLITHIC STRUCTURAL PARTSMONOLITHIC STRUCTURAL PARTS

••To Avoid joints (welding or riveting) To Avoid joints (welding or riveting)

••Homogeneous features Homogeneous features

••Good strengthGood strength--toto--weight ratio weight ratio

••In some cases high buyIn some cases high buy--toto--fly ratiosfly ratios

Boeing, 2003




20052005



Sometimes there is Sometimes there is a frame remaininga frame remaining Chips mean up to 95% in Chips mean up to 95% in

Weight!!!Weight!!!

Aluminium PlatesAluminium Plates

Final Monolithic PartFinal Monolithic Part

Monolithic Parts

Introduction




20052005



Machining Problems

STATIC PROBLEMSDeformations of the Deformations of the workpiece during the workpiece during the machining due to the machining due to the

cutting forces. cutting forces.

Monolithic Parts

Low Rigidity Local Zones

Thin Walls Thin Floors

DYNAMIC PROBLEMS••SelfSelf--excited vibrations excited vibrations (Chatter)(Chatter)••Forced vibrations Forced vibrations

Introduction




20052005



Roughing Semi-finishing Finishing

Tool vibration Both of them Part vibration

Regenerative Chatter

0º180º

90º

Introduction

Difference of phaseDifference of phase




20052005



Solutions

The machining of low rigidity partsinvolves theorical and

experimental know-how

Introduction




20052005



‘Jump to jump’ toolpaths.

135

246

••Taking advantage of the rigidity of the uncut workpiece:Taking advantage of the rigidity of the uncut workpiece:

Practical Solutions: Thin Walls

••Reducing forces in the less rigid direction of the wall:Reducing forces in the less rigid direction of the wall:

Bigger stockHigher rigidityBigger forces

Compromise must be taken

Bigger stock before the machining to rigidize the wall

••Relieved Shank Tools, to avoid unexpected reRelieved Shank Tools, to avoid unexpected re--cutting due to the static or dynamic deformations.cutting due to the static or dynamic deformations.

Bigger tool corner radius

Introduction




20052005



‘Two Level Step’ Toolpath

Bigger stockHigher rigidityBigger forces

Minimize corner radius

Practical Solutions: Thin Webs••Taking advantage of the rigidity of the uncut workpiece:Taking advantage of the rigidity of the uncut workpiece:

••Reducing forces in the less rigid direction of the web:Reducing forces in the less rigid direction of the web:

••Vacuum fixturesVacuum fixtures

••Dividing plunging in several steps to avoid machining Dividing plunging in several steps to avoid machining with full radial immersionwith full radial immersion

••Reducing feed rate in corners to avoid severe chatter Reducing feed rate in corners to avoid severe chatter vibrations or selecting cutting conditions in the stability vibrations or selecting cutting conditions in the stability lobes diagram taking into account that the radial lobes diagram taking into account that the radial immersion is bigger in cornersimmersion is bigger in corners

Introduction




20052005



•Aluminium 7075-T6

•300x50x0.3 (mm) thin walls

•Previous wall width (finishing ae): 1 to 8 mm

•Tool: 16 mm, r 2.5 mm, z 2, relived shank

•‘jump to jump’ toolpath

•24000 rpm, 3600 mm/min, ap 5 mm, ae 6 mm

Introduction

•Aluminium 7075-T6

•300x115 (mm) thin floor.

•Tool: 16 mm, r 0 - 2.5 mm

•0.5mm thickness floors achieved with vacuum fixtures.

•2 mm thickness floor successfully machined without backplate support.

Spiral Toolpath ‘Two Level Step’ Toolpath




20052005



Thin walls

Global ‘stability lobes diagram’, taking into account the relative displacement between tool and part.

A 3D model may be necessary because the web vibrates mainly in the tool axis direction (z) and the tool does it mainly in x and y directions.

Thin floors

Step 1

Step 3

Part features change along the milling.

A time ‘stability lobes diagram’ is necessary. At least a diagram with discrete-time steps.

Step 2

Step 4

Theoretical Solution: Calculation of the Stability Lobes

Introduction




20052005



IMC22IMC22

22nd International 22nd International ManufacturingManufacturing ConferenceConference

31st August-2nd September 2005

AvoidingAvoiding VibrationVibration in in thethe MillingMilling of of LowLowRigidityRigidity ComponentsComponents

Three Dimensional Stability Model

Why a tridimensional stability model?

A 3D stability model is not necesary when machining webs with a zero corner radius toolsince there is no regeneration of the chip thickness in the tool axis direction.

However, a fillet or a radius between thin webs and thin floors is usually required, so a bull-nose end mill is going to be used.

There will be cases when the use of a zero corner radius tool for pocketing and a bull-nose end mill for the fillet is more productive and cases when is better to machine all theworkpiece with a bullnose end mill.

To estimate the productivity, cutting conditions must be selected bearing in mind stabilitylobes, so a bidimensional and a tridimensional model is required.

Tridimensional models could also allow machining more complex surfaces with an end mill.

Two conditions are needed to apply a tridimensional stability model:

Workpiece flexibility in the tool axis direction

Regeneration of the chip thickness in the tool axis direction

Tridimensional models can also be applied to machine tool chatter, or fixtures chatter whenmachining steel or titanium at low spindle speeds.

We want to avoid fixtures




20052005




-+

-+

Intended chip thickness

Dynamic chip thickness

Block Diagram of Chatter Dynamics

Real chip thickness K*a

Cutting forcesFrequencyResponse Function

of theSystem

Dynamicdisplacement

Dynamicdisplacement

Dynamicdisplacement

in the lastperiod of cut

-Tse




20052005




X

φj

Ωκ

γ∆xsinφj+∆ycosφj

[∆xsinφj+∆ycosφj]sinκ

∆z

∆zcosκ

YZ∆y

∆x

( ) ( )[ ] ( ) ( )γγφφφ cossincossin)( ⋅∆+⋅⋅∆+⋅∆= zyxh jjjj

Chip Thickness Calculation

Intended chip thickness: Due to the feed per tooth (fz)sin sinz jf φ γ⋅ ⋅

Dynamic chip thickness:

Considers the displacements (due to vibration) of the cutting edge

( ) :j tφ Angular position of the cutting edge j

:κ Cutting edge lead angle.

:γ Axial immersion angle, complementary to κ

.




20052005




-+

-+






of theSystem

Dynamicdisplacement

Dynamicdisplacement

Dynamicdisplacement


-Tse




20052005




Cutting Forces Calculation

γFa

Fr γ

φj

Ft

Y

Xsin cosr aF Fγ γ⋅ + ⋅

Z

( )( )( )

( )'

'

1t

r t r j j

a a

F jF j K a K hF j K

φ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= ⋅ ⋅ ⋅⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪

⎩ ⎭⎩ ⎭

( )( )( )

( )( )( )

cos sin sin cos sinsin sin cos cos cos

0 cos sin

x j j j t

y j j j r

z a

F j F jF j F jF j F j

φ γ φ γ φφ γ φ γ φ

γ γ

− − ⋅ ⋅⎧ ⎫ ⎧ ⎫⎡ ⎤⎪ ⎪ ⎪ ⎪⎢ ⎥= − ⋅ ⋅ ⋅⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥− −⎣ ⎦⎩ ⎭ ⎩ ⎭

Mechanistic forces model(Altintas)

Tangential, radial and axial cutting coefficients obtained troughexperimental tests measuringforces with a Kistler dynamometerand adjusting those coefficients in a way that the modelled forcesmatch the measured ones.

Depth of cut




20052005




Cutting Forces Calculation

( ) ( ) ( ) 12 tF t a K A t t= ⋅ ⋅ ⋅ ⋅ ∆⎡ ⎤⎣ ⎦

Forces and displacements related by Forces and displacements related by the matrix of directional coefficients: the matrix of directional coefficients: ( )A t⎡ ⎤⎣ ⎦




20052005




-+

-+






of theSystem

Dynamicdisplacement

Dynamicdisplacement

Dynamicdisplacement


-Tse




20052005




Frequency Response Function of the System

( )[ ]( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

cwcwcw

cwcwcw

cwcwcw

ctctct

ctctct

ctctct

c

iGzziGzyiGzxiGyziGyyiGyxiGxziGxyiGxx

iGzziGzyiGzxiGyziGyyiGyxiGxziGxyiGxx

iGωωωωωωωωω

ωωωωωωωωω

ω

In a general case, In a general case, FRFsFRFs of both the tool and workpiece must be considered.of both the tool and workpiece must be considered.

In rough milling or roughing and finishing if the In rough milling or roughing and finishing if the workpiece is rigid enoughworkpiece is rigid enough

Finishing if the workpiece is very flexibleFinishing if the workpiece is very flexible

Only the FRF of the tool is Only the FRF of the tool is accurate enoughaccurate enoughOnly the FRF of the workpiece Only the FRF of the workpiece is accurate enoughis accurate enough




20052005




-+

-+






of theSystem

Dynamicdisplacement

Dynamicdisplacement

Dynamicdisplacement


ecTω




20052005




Stability Equation

From the block diagram, the response in frequency domain is:From the block diagram, the response in frequency domain is:

( ) ( ) 1 c ci T i tce G i F eω ωω−∆ = − ⋅ ⋅⎡ ⎤⎣ ⎦

So:So:

( ) ( )[ ] ( )[ ] tict

ti ctci

c eFiGtAeKaeF ωω ωω

⋅⋅⋅⋅−⋅⋅⋅=⋅ −121

To solve this equation as an To solve this equation as an eigenvalueeigenvalue problem must be invariable in time:problem must be invariable in time:( )A t⎡ ⎤⎣ ⎦MonofrequencyMonofrequency solution:solution:

MultifrequencyMultifrequency solution:solution:

Mean term of the Fourier series expansion of Mean term of the Fourier series expansion of is consideredis considered

Several terms of the Fourier series expansion of Several terms of the Fourier series expansion of are consideredare considered

( )A t⎡ ⎤⎣ ⎦

( )A t⎡ ⎤⎣ ⎦

Less accuracy in high interrupted cutting

Less time consumingMore time consuming




20052005




Stability Equation

( ) [ ] ( ) 01 12

c c ci t i T i tt cF e a K e A G i F eω ω ωω−= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅⎡ ⎤⎣ ⎦

[ ] ( ) [ ] ( )01det 1 02

ci Tt cI a K e A G iω ω−⎡ ⎤− ⋅ ⋅ ⋅ − ⋅ ⋅ =⎡ ⎤⎣ ⎦⎢ ⎥⎣ ⎦

[ ] ( )0det 0cI G iω⎡ ⎤+ Λ ⋅ =⎡ ⎤⎣ ⎦⎣ ⎦ ( )12

ci Tt

N a K e ω

π−Λ = − ⋅ ⋅ ⋅ −

A cubic function is obtained, solving the problem analyticallyA cubic function is obtained, solving the problem analytically

It is an It is an eigenvalueeigenvalue problem, so:problem, so:

For an array of values of “possible frequencies of chatter”, For an array of values of “possible frequencies of chatter”, eigenvalueseigenvalues are are obtained and also the corresponding depths of cut and spindle spobtained and also the corresponding depths of cut and spindle speedeedss




20052005




Stability Equation

( )( )

tan tan2 2

creal Timag

ωπκ ψΛ ⎡ ⎤⎛ ⎞= = = − ⎜ ⎟⎢ ⎥Λ ⎝ ⎠⎣ ⎦

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

1

1

1

1

0, 0 tan

0, 0 tan

0, 0 tan

0, 0 2 tan

imagreal imag

real

imagreal imag

real

imagreal imag

real

imagreal imag

real

ψ

ψ π

ψ π

ψ π

−

−

−

−

ΛΛ > Λ < → = −

Λ

ΛΛ < Λ < → = − +

Λ

ΛΛ < Λ > → = − −

Λ

ΛΛ > Λ > → = − +

Λ

2cT kω ε π= + ⋅

22 2

2 2 , para que 2 02

k

k k

π ε πψ

ε π ψ π ε πε π ψ

+⎛ ⎞= − ⎜ ⎟⎝ ⎠

= − + < → == −

( )1 2c

T kε πω

= ⋅ +( )6060

2cn

N T N kω

ε π⋅

= =⋅ ⋅ +

( ) 2lim

2 1t

a realN Kπ κ⎡ ⎤= − ⋅ Λ ⋅ +⎣ ⎦⋅

EigenvaluesEigenvalues

Depth of cutDepth of cut

Phase of the Phase of the eigenvalueeigenvalue

Difference of phase in the Difference of phase in the undulation left by the tool undulation left by the tool between two periods of cutbetween two periods of cut

Possible Possible frequencies of frequencies of chatterchatter

Period of cut for each lobe k (k=0,1,2,3…)Period of cut for each lobe k (k=0,1,2,3…)

Spindle speed for each lobe k (k=0,1,2,3…)Spindle speed for each lobe k (k=0,1,2,3…)

0º180º90º

k (k=0,1,2,3…) represents the number of complete undulations k (k=0,1,2,3…) represents the number of complete undulations between two periods of cutbetween two periods of cut




20052005




Considerations about bull nose end mills and ball end mills

γγ

a a

The axial immersion angle is variable along the depth ofThe axial immersion angle is variable along the depth of cutcut A constant value is needed to solve A constant value is needed to solve the problem analyticallythe problem analyticallyγ

Also, due to helix angle variation along the tool axis directionAlso, due to helix angle variation along the tool axis direction, cutting coefficients vary. , cutting coefficients vary.

174,0248,26+52,15*akac(N/mm2)

804,731723,3-369,44*aktc(N/mm2)

65,495558,35-198,93*akrc(N/mm2)

Frontal partToroidal part

SolutionsSolutions

AveragingAveraging

IterateIterate

Average value of for the depth of Average value of for the depth of cut that is expected to be cutcut that is expected to be cut

γ

Obtain stability lobes for several Obtain stability lobes for several ranges of the depth of cutranges of the depth of cut

42ππγ +=

Ex: For a depth of cut equal to Ex: For a depth of cut equal to the tool corner radiusthe tool corner radius




20052005



Experimental Validation

••Straight cuts in Aluminum 7075T6Straight cuts in Aluminum 7075T6

5880805,220,010834,69

1718199,550,010728,24Piece ZZ

15909208,470,071651,23

19297178,140,086373,20Tool XX, YY

Stiffness(N/m)Damping(%)Frequency(Hz)

••Four springs of K=312 N/mm below Four springs of K=312 N/mm below the steel platform and two of K=189 the steel platform and two of K=189 N/mm above the platformN/mm above the platform

••Bull nose end tool Bull nose end tool KenduKendu KENAL KENAL 4400.60 with two cutting edges,16 mm. 4400.60 with two cutting edges,16 mm. diameter and 2.5mm radiusdiameter and 2.5mm radius

••Radial immersion of 50%Radial immersion of 50%

••Vibration measured by accelerometers and a signal Vibration measured by accelerometers and a signal analyzer OROS OR35.analyzer OROS OR35.

Unstable cut (a=0,5 mm, n=900 rpm)

Stable cut (a=0,5 mm, n=2000 rpm)

•γ Averaged for a depth of cut= 0.5 mmAveraged for a depth of cut= 0.5 mm

FORCED VIBRATIONS!!!!!




20052005



Experimental Validation

CHATTER

CHATTER

STABLE

FORCED VIBRATION




20052005



Conclusions

The finish milling of a low rigidity structural part can result in dynamic problems limiting the productivity and the quality of the workpiece so it is necessary to stiffen the part with fixtures, which can be an expensive solution, or select the an optimal set of cutting conditions based on the workpiece dynamics.

A three dimensional stability model has been developed bearing in mind the machining of thin walls and thin webs. A validation device with invariable dynamic properties has been used to validate the model in order to avoid the uncertainty introduced by the variable dynamic behaviour of a thin web in a real case.

The model has been applied also for the case of ball end mills and bull nose end mills with a variable cutting edge lead angle, solving the non linearity introduced by averaging the cutting edge lead angle and the cutting coefficients.

In order to improve the milling process planning, care must be taken with the forced vibrations since the most stable areas in the lobes diagram match the modal frequency or a multiple and amplification of the vibrations may occur, resulting in a higher depth of cut that leads to instability or an overcut or undercut of material.

Dpt. Of Mechanical Engineering Primer Congreso ...ehu.eus/manufacturing/docpubli/200.pdf · Dpt. Of...

Documents

Transcript of Dpt. Of Mechanical Engineering Primer Congreso ...ehu.eus/manufacturing/docpubli/200.pdf · Dpt. Of...