A Multivariable Simulation Model for the Dynamic Behaviour of a Machining Center During Milling Operations

Copyright Cl IFAC Intelligent Manufaauring Systems. Vienna. Austria. 1994

A MULTIV ARIABLE SIMULATION MODEL FOR THE DYNAMIC BEHA VIOUR OF A MAClDNING CENlER DURING MIIJ,ING OPERATIONS R Koorad, R. Isennarm Technical Univcmty of DaJmstadt, Institute of Automatic Control, Laboratory of Control Engineering and Process Automation. I..andgI3f-
Abstract. An entire model for a machining center during milling operations is presented. Existing and partly improved models of the milling process and the respective machine components are coupled together. Because of the modular structure single components can easily be exchanged and the entire model can be adapted to various types of machine tools. Simulation results show the usability to predict interactions in the drives caused by the intermittent cut. These interactions are significantly influenced by faults in the milling process, e.g. tool breakage and cutter run-out. Keywords. Computer simulation, cutting, electric drives, machine tools, modelling, multi variable systems I . INTRODUCTION

The model structure and selection of components (e.g. types of drives) are adapted to a MAHO MC 5 machining center existing at the Institute of Automatic Control at the Technical University of Darmstadt.

During milling operations the resulting cutting forces cause interactions in the different feed drives and the main drive. The intermittent cut excites vibrations in the machine tool structure and consequently the cutting forces in turn are affected, see also Tarng et al. (1994). The milling forces are mainly dependent on the penetration between the cutter inserts and the workpiece. Because this penetration is a function of the main spindle speed and the speed of the feed drives a multivariable model is reasonable. In this contribution the simplified case of linear motions in x-feed direction is considered. An entire model is developed which can be used to simulate the interactions in the drives by coupling already known and partly improved models of the milling process and the respective machine components. This model can be used for model based fault diagnosis of the milling process, working with simply measurable signals, like motor currents, see also Isermann (1993).

2.2 Model of the milling process The force effecting the milling tool is the geometrical sum of all single forces effecting the inserts being actually in penetration with the workpiece. These single forces can be computed based on the chip cross sectional area, which is the product of the depth of cut and the chip thickness (IGm and Ehrnann, 1992). Starting out from this static force model and under the assumption that no axial runouts of the cutting inserts occur a new method for the calculation of the chip thickness is derived in the following. This method is usable for variable main spindle speed, variable feed speed and radial run-out of the inserts. Basis of the method is the description of the milling tool position in the workpiece coordinate system with reference to the lower left corner of the workpiece surface to be machined, see Fig. 2.

2. MODELLING

2.1 Modular structure of the model I

Because of the complexity of a whole machine tool a modular structure of the simulation model is reasonable. The modules are coupled via flow and effort variables. Flow variables are accelerations and effort variables are forces , respectively torques, see Fig. I.

(x2,y2) (x4.y4)

r I

I

(xl,y:)__ _

Module 1

14-1.--,:..:.::...-----i'1 Module

Fig. 1 Scheme of oonnection of

2

(xS,yS)

x

Fie. 1 Posinon of cutter in wodtpiece ~stcrn

two IOIalory moving moduIcs (eg.

gearbox-shaft)

The axis of the cutter has the coordinates (x.(k),y.(k». The position of the cutting edges (Xj(k),Yj(k» are defmed by

309

~t=t(k)-t(k-1).

Under the simplification to linear course the new values of x.. lie on the straight line defmed by x=ay+b with

cp fk) =cp (k) +(i -1) 27t {\:

%

I

index of tooth i=l..z nuniler of teeth angular position of spindle angular position of tooth I (
with

z
r;

k

=
,....'''27\

,,

,

", \

"

I

.

Of cause, linear interpolation is an approximation which is the more inaccurate the bigger the distance between two simulated positions. Experiments have shown that performing simulations in the time domain the simulation step time has to be decreased with increasing cutting speed, which is a function of spindle speed and tool radius.

(D,yi)

The one-dimensional description of the border line provides a powerful, but relatively simple method to consider radial run-out of the cutting edges. In extreme cases the run-out of the previous tooth is such big that the actual tooth stays left of x.., i.e. no contact happens. In analogy x.. remains unchanged if a tooth is broken. In that case the following tooth has the corresponding higher load . Another advantage of the described method is that the simulation of ent:')' transients is feasible without further differentiations. In this case the initial cutter axis position (x~ YAl) has to be chosen such, that the cutter is completely out of the workpiece.

' .. " <... - ,

,

7

, , ' XIra(O )

ofx., in cbmgedpct

DeW c:ourwe

Case I: tooth i enters wodcpicce. Case 2: tooth i remains in workpiece. Case 3: tooth i leaves workpicce

I I

, ......

t . . . . . . .(

.....0

Fig. 4 Calcularion of new coun;e of border line between machined and unmachined surface for time interval At;::t(k}t(k-I).

- - - -1Z,i

:~) ~ ','

,

old c:ourwe of x., in cbmgedpct

.

®

-..(10)

~-

coune of tooth i

~?_.'_.'"

(xi(k),yi(k)~r

1. Storage of the border line between machined and unmachined surface in a vector x..' where x.. is the xcoordinate and the index of x.. is the y-coordinate of the border line. In general for the milling operation a scale in ~ is reasonable. The length of x.. results from the height of the workpiece Yo in ~. Assuming that the workpiece is still unmachined x.. is equivalent to the left edge of the workpiece, i.e. for a rectangular surface x..=Q. see Fig. 3a.

~

c:::,'....'_.'"

~xi(k),yi(k» ® ~xi(k),yi(k»


In detail the chip thickness is evaluated the following way:

~ -~- "

b=x(k)-ay(k)

Because the cutting edge of tooth i can enter or leave the workpiece during one simulation interval. the range of interval in which x.. is updated can be smaller than the range between the simulated cutter positions (xi(k-l),Yi(k1» and (X;(k)'Yi(k». Several cases have to be distinguished. see Fig. 4.

The calculation of the chip thickness is performed by determination of the cutting depth in x-direction for each tooth separately. This cutting depth in x-direction results as a difference between the actual x-position of the respective cutting edge and the x-coordinate of that point that lies on the border line between machined and unmachined surface and has the same y-coordinate.

,,.,

twI

a=x(k)-x(k-l) )'(k)-)'(k-l)

"

Hg. 3 Description of border line between machined and unmachined surface by an one-
2. The cutting depth of tooth i in x-direction which is also called the actual feed fzj(k) of tooth i is

The presented method has its limitations if situations have to be simulated where the cutting edges leave the workpiece on the right side, e.g. because of interrupted surface geometry or exit transients. In these cases a twodimensional scale of the workpiece surface would be necessary.

(see also Fig. 3b). The actual cutting depth h;(k) of tooth i is found from the equation:

h,(k)=/Jk)sin(cp,(k)-7t) After the calculation of h;(k) the force effecting tooth i can be divided into tangential, radial and axial components determined by the equations given by Konig et al. (1981) :

3. If at the actual simulation time t(k) tooth i is in penetration or has left the workpiece since the last simulation time t(k-l). x.. has to be updated with regard to the course of the edge during the respective time interval

F fIC) =a k d'

310

p d,l

sinx --Ch fIC)I--c {\

A transformation into the workpiece coordinate system is performed by x.(Ic)=x~{Ic) +xttfJ

with Fa F, Fp;

with N N N

a,.

mm

Jr.,., Icn.,

N'rm1' N'rm1'

k" ., Nlrmr l-ro.

1-fIlt

l-m"

h; x

Degree

ClJtting fon:e of tooth i Feed fon:e of tooth i Passive fon:e of tooth i Back engagement Specific cutting fon:e unit Specific feed fon:e unit Specific passive fon:e unit The rise in rutting fon:e related to the nominaJ width of rut as a function of the nominaJ thickness of rut The rise in feed fon:e relamd to the nominal width of rut as a function of the nominaJ thickness of rut The rise in passive fon:e related to the nominal width of rut as a function of the nominal thickness of rut ThiclcnessofOu Tool cutting edge angle

X,

x,.,

H:oordinare of the rutt.er axis Initial x coordinate of the rutt.er axis

In the block 'berechne Kriifte und Momente' the calculation of the cutting forces and torque is performed. In addition to the SIMULINK-model, a list with all relevant cutting parameters exists.

2.3 Model of the main drive The main drive of the investigated machining center consists of a DC-motor with flux weakening and the mechanical components belt-drive, shaft, gearbox and spindle. A speed-controller with cascaded control loops for armature current, armature voltage and exciting current is implemented. The resulting structure for the main drive is shown in Fig. 6. For recording during simulation some inner states are drawn out of the blocks.

Decomposition yields the force components effecting the workpiece

, , I: F d{Ic)costp~Ic)-I: F J.1c)sintp~lc) '-I '-I , , F,{Ic) =I: F JIc)sintp~Ic)-I: FJ.Ic)costp~lc) '-I '-I , F,{Ic) =I: F,tic) '-I Filc)=-

f ......... '

Fig. 6 SIMULINK rc.alizabon of rnain-drive

The torque effecting the spindle is given by

The speed controller has PI-structure with limitation of the output variable which is the set point for the armature current The speed controller contains an additional input for the external stop of the integrator track. This is required if a controller in a cascaded control loop is in limitation. The controller for the armature current IAIlhas PI-structure with limitation of the output variable, which is the set point of the armature voltage U All' Because of the thyristor bridge the armature voltage is delayed, which is modelled with a dead-time element. If the armature voltage reaches its maximum value the voltage controller inhibits a further increase and produces a decrease of the exciting current lE causing the flux weakening. The electrical part of the motor is given by

,

I: r ,.JIc) '-I

M{Ic) =

The realisation of the milling model with SIMULINK is shown in Fig. 5 (The Math Works, 1993).

To WorklPIC.:

""chnlKr'" undMom,,*

M...

YJg. 5 SIMULINK rc.alizabon of milling proca;s with qv.

The angular acceleration of the driving spindle tpSH is integrated twice

Motor position R"" Annatwc resistance L"" Annatwc inductancc 'f' H magnetic flux

The mechanical part is detennined by In analogy, the table position x. is given by

with J...

Mu. 311

rronatt of Inertia tm:hanical load

2.5 St7Ucture of the entire system In the model Coulomb behavior is assumed for the sliding friction. In addition. in view of start and stop events peak stick friction is implemented to consider idle states of the components. Belt drive and gearbox of the drive are modelled as rotatory mass-spring-damper systems with friction and gear ratio. In comparison. shaft and spindle are considered stiff. A derivation of the differential equations for such systems can be found in Isermann et al. (1991). If. as in the regarded case. two mass-damper-spring-systems are coupled. the states at the load side of the first system are identical with those on the driving side of the second system. The respective moments of inertia have to be added. The resulting entire inertia is then assigned to the first system.

The entire system is yield by linkage of the drives and the milling process. After grouping to the block ' prozeB' the structure in Fig. 8 results. The inputs ' Solldrehzahl HAMotor' and 'X-Sollposition Tisch' contain the courses of the set points for main drive motor speed and table position. Vector x contains selected variables of the process. All physical parameters of the drives are stored in a separate list These parameters were obtained using data sheets. physical laws or identification methods (Reiss. 1992; Wanke. 1993).

~

Clock

Simulationszen

1!1 ;phlP_LMsoIIHJ' ~~--+I SolldrahZahl H.lrMotor

2.4 Model of the x-feed drive The x-feed drive consists of a permanent excited synchronous motor and the mechanical components belt drive. threaded spindle and slide with table. The table position controller has P-struCtllre. With a PI-behaviour with lag for the cascaded speed controller the feed drive has the structure shown in Fig. 7.

To VVorkspace X-Sollposition TIsch

Fig. 8 SimuIink realization of eoIire model

3. SIMULATION RESULTS

3.1 Example 1: Simulation with variable feed per tooth Example 1 will prove the usability of the model for simulation with variable spindle and feed velocity and also for entry transients. Therefore a milling operation was simulated under the following cutting conditions:

Fig. 7 SIMULINK realization of x-feed..drive

In combination with the current control the motor is modelled by a first order system

Wodcpiece Spindle ipeCd Set point of feedhooIh Milling type DepIh of all Number of inserts Tool radii heigb1 of workpiece iniliaI pas. of aJIIec axis sampling time simu1ation time aIlIing edge ang1e

iNTUb+IN-I. wilh

I..

I" T_

of all IDOIOr currerus &et point for cum:nl control time ronsIaDl of \be cum:nl control SlDD

The mechanical part is given by

wiIh

cpy. J..

Mu

c,., = 0.2.0.25..0.2 IDII end milling 1 IDII

3 r, = r2 = r, = 0.025 m Y.= 0.03 m x., 0.0255 ID, Ym 0.015 m

=-

=

0.4 MS 1.7 s le

= 90"

At the beginning the cutter is completely outside of the workpiece. at t \=0.01 s the main drive is started with a spindle speed setpoint of 1200 rpm. At t 2=0.4 s the feed drive is started with a speed of 12 mm/s, which results in a setpoint for the feed per tooth f J!J of 0.2 mm. After appr. 0.5 s the cutter touches the workpiece edge. After appr. 0.9 s the entry transient is over and the cutting process has reached its stationary phase. At t r1.l s the spindle speed is reduced to 80 % which causes an increase of f J!J to 0.25 mm. At t.=1.4 s the feed velocity is also reduced to 80 % with the result that f J!J is again 0.2 mm. Some of the

Mocor posiIion of Iner\ia mecbanicalload

IDOIJIeIlI

~ friaion

'1'.

Steel ST50 1200..960 qm

in \be IJIIlIOr besings magnetic flux

For the belt drive the same model is used as for the main drive. Threaded spindle. slide and table can be described with a mass-damper-spring system with a transmission of rotatory into translatory motion. see Isermann et al. (1991).

312

resulting variables (forces. torques. positions. currents .. ) are shown in Fig. 9 to Fig. 11. The interactions in the drives caused by the cutting process are obvious. As Fig. 11 shows every single tooth engagement can be seen in the annature currents of both drives what gives motivation to use the currents for diagnosis purposes in a further step.

i j~7~:-~

w~

rrein riiw

p/I",- -

o

.

0.2

0."

1.2

1.4

IUS

1.4 s I( = 90"

0.2

0.'

After appr. 0.7 s the cutting process has reached is stationary phase. At t\=1 s the breakage of tooth 5 is simulated by setting r~. Some of the resulting variables are shown in Fig. 12 and Fig. 13 for the time period within the breakage happens. Before the breakage the runout causes a typical force pattern. which are similar to those shown by Tarng and Lee (1992). As expected. the breakage causes a significant change in the force pattern and in the resulting drive variables.

0.0

0.8

1.2

1.4

1.0

1 .-teed c;;.,.

O.OIL~~~~~-;";,,-ue! ,

E a.cost

I_I

0.2

0."

0.8

0.8

1.2

1.4

1.6

1

0<

0.4

1.0

v;,

;'g1C~

0

00

r, =r, =r. =r" =0.0"...5 m r, = 0.02505: r, = 0.025 ..0 yo= 0.03 m x., =- 0.0255 m. y .. =0.015 m

1

~:t=: 'CO > 0

0.8

0.'

=

beigbl of wodqliece iniL pas. of CUller axis sampling time simula1ion time cutting edge angle

400

¥

Sleel ST50 1200 qm c,., 0.2 mm end milling Imm 6

Spiodle speed Set point of feedllooCh Milling type IlqlIh of an Number of insens Tool radii

It'--l-":--'-':~""wm.:"'1'_'J.'''-=

t

10:r

OL---0~.2~~O. '~~O~ . O--~O~ . O--~--~'.~2--~I~ . '--~I~.O~

l~~~~~

I

--

Fig. 9 SimulaJed drive signals foc varying spiodle md 1ab1e vekx:ily

8.8

0 .15

0.8

0.05

1.05

t

torc.ln.~

0.2

0.4

0.'

0.8

1.1

us

1.2

F_x

.~

12 1

trcwc.1n • ......".

-ZOS..

z

0.15

0.8

O.ts

1

t ot

1.05

US

1.2

I

Fig. 12 Spiodle LDrqUe IOd fon:e in x-direc:lion for axial nmOUl md 1001 breakage at 1=1 s (1=0.8 .. 1.2 s)

o<':~

HI!. 10 SimuIaJed spindle LDrqUe IOd fon:e in x-direc:lion __ M

8..

Cl Cl Cl Cl rl 1.32

1.304

1.38

1.»

1.34

1.3e

1.31

1.38

8.8

1.»

1.1

1.15

1.2

0.'

0.15

1

1.05

1.1

1.15

1.2

o<':~

1,4

o<':~ 1.34

1.05

I

I

1.32

1

tMd ci'tw ermat\n curNft NIt potnt I_a

mein dnrIfre ItnNtlle CWTMII_AM

~.3

0.15

I __ """,_SH

0 .15

zso:p~~ 1..3.2

0.1

j::~'~"'~M~~~ 2'8..

1.4

tOfCe in .-dirCion F_x

-~ .3

0.85

0.15

0.1

0.85

1 I x-4Md v.Iodty

l.OS

1.1

1.15

1.2

1.1

1.15

1.2

1;)_-

0 ..... _

1

0 .02•

1."

o.~..

I

teed cnw. anNhn C\nWIt ... point 1_o

0.15

0.1

O.ts

1

1.OS

I

HI!. 13 Selection of drive signals foc axial nmout and 1001 breakage at 1= 1 s (1=0.8..1.2 s) HI!. 11 Seleclion of cutting me! drive signals (1=1.3 .. 1.4 s)

4. CONCLUSIONS In this contribution different already existing models for the components of a machine tool and for the milling process were coupled to an entire model. Especially the milling model had to be improved in view of varying spindle and feed velocities. Using SIMULINK a simulation software was developed which is usable for the simulation of milling operations with motions in one feed direction. Because of the modular structure the model can easily

3.2 &le 2: Simulation of cutter run-out and tool breakage In order to prove the usability of the model for the simulation of radial cutter run-out and tool breakage the cutting conditions were changed as follows:

313

adapted to other machine types by rep1acement of single components. The control structure can also be easily modified, e.g. to test strategies for force control. In future the simulation results have to be verified with measurements on the test stand. Further investigations of the relation between faults in the cutting process and the resulting drive variables have to be performed in view of a model based fault diagnosis system using those simply measurable variables.

5. REFERENCES

Isermann, R. et al. (1991). Model based fault diagnosis of machine tools. 30th Conference on Decision and Control. Brighton. Isermann R. (1993). Fault diagnosis of machines via parameter estimation and knowledge processing, Automatica, 32, No.4, 815-835. Kim, H.S., Ehmann, K.F. (1992). A cutting force model for face milling operations. IntemationalJournal of Machine Tools and Manu/acturing,33, No.5, 651-673. Konig, W. et al. (1981). SpeziflSche Schnittkraftwerte flir die Zerspanung metallischer Werkstoffe. Verein Deutscher Eisenhilttenwerke. Verlag Stahleisen. Dilsselorf. Reiss, T. (1992). Fehlerfriiherkennung an Bearbeitungszentren roit den MeBsignalen des Vorschubantriebs. Fortschr.-Ber. VDI Reihe 2 Nr. 286. Diisseldorf. Tamg, Y.S. et al. (1994). An analytical model of chatter vibration in metal cutting. International Journal of Machine Tools and Manufacturing,34, No.2, 183-197. Tamg, Y.S., Lee, B.Y. (1992). Use of model-based cutting simulation system for tool breakage monitoring in milling. International Journal of Machine Tools and Manufacturing,32, No.5, 641-649. The Math Works, Inc. (1993). Matlab User's Guide. Massachusetts. Wanke, P. (1993). Modellgestiitzte Fehlerfriiherkennung am Hauptantrieb von Bearbeitungszentren. Fortschr.-Ber. VDI Reihe 2 Nr. 291. Dilsseldorf.

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