contributions regarding the constructive and...

"LUCIAN BLAGA "UNIVERSITY OF SIBIU

FACULTY OF ENGINEERING

Eng. CRISTINA MARIA BIRIŞ

CONTRIBUTIONS REGARDING THE

CONSTRUCTIVE AND FUNCTIONAL

OPTIMISATION OF NUMERICAL

CONTROL CUTTING SYSTEMS

- ABSTRACT OF THE PH.D. THESIS -

PH.D. ADVISOR

PROF. OCTAVIAN BOLOGA, PH.D.

2008

2

CONTENTS

Introduction 5 5

1. State of the art in the domain of NC cutting systems 7 9

1.1 Modern cutting procedures 7 9

1.1.1 Generalities 9

1.1.2 Thermal cutting by burning the metal 11

1.1.3 Thermal cutting by melting the metal 13

1.1.3.1 Plasma jet cutting 13

1.1.3.2 Laser cutting 16

1.1.3.3 Arc cutting 18

1.1.4 Other cutting procedures 19

1.1.4.1 Waterjet and abrasive jet cutting 19

1.1.4.2 Electron beam cutting 21

1.1.4.3 Cutting by electro-discharge machining 22

1.1.5 Critical analysis of the various cutting systems 23

1.2 Structural analysis of the NC cutting machines 11 26

1.3 Motion control systems used in cutting machines. Numerical axes 12 29


1.3.2 Numerical axes 31

1.4 Aspects of the analysis of automated systems from NC machines 14 37


1.4.2 Defining the transfer functions of automated systems 40

1.5 Study of the dynamic behaviour and of the kinematic precision in

NC cutting machines 15 48

1.5.1 Disturbance factors in the functioning of kinematic feed chains 48

1.5.2 Optimisation of the dynamic behaviour by using controllers 51

1.5.2.1. The PID controller 51

1.5.2.2. The fuzzy controller 52

1.5.2.3. The feed-forward controller 53

1.5.2.4. The cross-coupling controllers 54

1.5.2.5. Adaptive controllers 55

1.6 Preliminary conclusions and objectives of the thesis 16 56

1.6.1 Conclusions regarding the state of the art in the domain 56

1.6.2 Objectives of the Ph.D. thesis 59

3

2. The mathematical model of a feed drive from the structure of

numerical-control cutting machines 18 61

2.1 General considerations regarding the mathematical modelling 18 61

2.2 Structure of the studied numerical axis 18 62

2.3 Determining of the relationships that characterise the functioning

of the numerical axis 19 64

2.4 Analysis of the kinematic precision of the numerical axis 21 72

2.4.1 Characterisation of the positioning regime 75

2.4.2 Characterisation of the contouring regime 78

3. Study by simulation of the behaviour of a feed drive from the structure of

numerical-control cutting machines 25 82

3.1 Generalities regarding the analysis by simulation of numerical axes 25 82

3.2 Structure and characteristic parameters of the studied numerical axis 25 84

3.3 Study by simulation of the dynamic behaviour in positioning regime 27 90

3.3.1. Generation of the reference kinematic parameters 90

3.3.2. Carrying out the simulation and interpretation of the results 92

3.3.3. Tuning of the position controller 95

3.4 Study by simulation of the generation, by cutting, of 90° corners 30 105

3.5 Study by simulation of the generation, by cutting, of circular contours 33 117

4. Experimental researches regarding the optimisation of the dynamic behaviour of

kinematic feed chains from the structure of numerical control cutting machines 36 126

4.1 Equipments used for experimental researches 36 126

4.1.1 The oxy-gas cutting machine OXYTOME HPC 20 126

4.1.2 The laser cutting machine MAZAK NT-X48 Champion 128

4.1.3 The numerical control unit FAGOR 8055 131

4.1.4 The coordinate measuring machine ZEISS PRISMO 7 S-ACC 132

4.1.5 The measuring machine ZEISS CONTOURECORD 1600D 134

4.2 The material used and the parts realised for the experimental researches 37 136

4.3 Studies concerning the influence of the cutting regime on the mechanical

and technological characteristics 38 139

4.3.1 Determining the surface roughness 139

4.3.1.1 General theoretical considerations 139

4.3.1.2 The equipment used for determining the roughness 139

4.3.1.3 Influence factors and determining the experimenting conditions 141

4.3.1.4 Statistical processing of the experimental data 146

4.3.1.5 Conclusions 149

4.3.2 Determining the microhardness 150

4.3.3 Determining the microstructure 153

4

4.4 Studies regarding the optimisation of processing by laser cutting on

NC machine-tools 41 155

4.4.1 Experimental study of generating, by cutting, of 900 corners 156

4.4.2 Experimental study of generating a circular profile 161

5. Final conclusions and considerations. Main contributions of the thesis 47 167

5.1 Final conclusions and considerations 47 167

5.2 Contributions of the thesis 50 170

5.3 Directions for continuing the researches 50 171

References 51 172

5

INTRODUCTION

The current Ph.D. thesis was realised in the period November 2005 – October

2008, within a full-time Ph.D. preparation stage, at the Faculty of Engineering of the

"Lucian Blaga" University of Sibiu.

The thesis' topic falls within the larger topical area related to optimising the

functioning of NC machine-tools, tackled by the members of the Research Centre

for Metal Forming from the Faculty of Engineering of the "Lucian Blaga"

University of Sibiu.

Currently, at world level, non-traditional technologies account for about 15-

20% of the total technologies applied, whereas for Romania the percentage is of only

3-4% [140]. With regard to the processing by cutting, the various non-traditional

cutting processes - with laser beam, plasma jet, water jet or abrasive jet, electro-

discharge machining etc., have come to basically hold a monopoly. Therefore, the

study of modern numerical control (NC) machine tools, used for carrying out cutting

operations, presents a justified interest.

The specific structure of these systems, especially the lack of a main kinematic chain

that would provide the rotation motion of the main shaft (which is replaced by the unit

carrying the processing beam), determines the processing precision to be decisively

influenced by the structure and characteristics of the feed kinematic chains.

The usage of an open NC architecture allows the modifying of regulation

parameters by the user in order to gain optimal performances. The current thesis aims

to contribute to clarifying several problems related to this aspect, as well as to create

useful theoretical and experimental instruments that could aid the user in realising the

above-mentioned process.

The formulation of the thesis' objectives was based on the conclusions drawn from

the analysis and synthesis of the state of the art of researches in the domain of cutting

processes and in the domain of feed motion control on NC cutting machines.

In order to achieve these objectives, the author has unfolded direct experimental

studies on NC cutting machines, but resorted also to a series of methods specific for

modern scientific research, such as mathematical modelling and numerical simulation

of motion control processes.

The thesis' structure takes in account, on the one hand, the specific elements of

the tackled domain and on the other hand the manner in which research activities were

drawn out in the author's doctoral programme, combining bibliographical researches

with detailed theoretical and experimental researches.

6

For the competent guidance, trust and support that he granted her throughout

the Ph.D. preparation period, the author wishes to express sincere thanks and deep

gratitude to her scientific advisor, Prof. Octavian Bologa, Ph.D.

Also, the author wishes to thank the members of the Commission for the

analysis and presentation of the Ph.D. thesis for their contribution in the phase of

thesis finalisation and evaluation.

The finishing of the thesis in such a short timeframe is also due to the support

which the author had in realising the theoretical and practical researches comprised in

the thesis, from the colleagues from the Faculty of Engineering of the "Lucian Blaga"

University of Sibiu, especially from those within the Machines and Equipment

Department, whom the author wishes to express here too warm thanks.

Feelings of gratitude are aimed also at the management of the company S.C.

COMPA S.A. Sibiu, who allowed the author the access to the oxy-gas cutting systems

and laser cutting systems, but also to coordinates measuring machines within the

company, all equipments without which a large part of the experimental researches

carried out would not have been possible.

A large part of the researches within this thesis were possible also thanks to the

financing provided by CNCSIS through the financing programme for researches for

young Ph.D. students type PN-II-RU-TD-2007, the author gaining through national

competition and successfully finalising the project TD-250 titled, like the thesis,

"Contributions regarding the constructive and functional optimisation of

numerical control cutting systems", contract no. 154/2007. Hereby the author wishes

to thank the CNCSIS personnel for the good cooperation during the entire period of

the project's unfolding.

Not least, the author wishes to express her love and profound gratitude for her

family, for the understanding and moral support provided in all these years, in which

the professional considerents often were put in first place.

7

Chapter 1

STATE OF THE ART IN THE DOMAIN OF

NC CUTTING SYSTEMS

1.1 Modern cutting procedures

The current industrial production is characterised by shorter and shorter design-

production cycles and by an increasingly larger variety of products that have to cover

the growing demands of consumers. The usage degree of sheet metal processing

technologies has steadily gained ground in the last years compared to other

technologies, because the processing of sheet metal gives the producers access to

design methods that are less restricted by rules and the possibility to manufacture parts

in a more simple, cheaper manner, while respecting the technical and quality

requirements of the beneficiaries.

The choice of a cutting procedure can be made function of the precision imposed

to the cut parts, of the material's hardness, its quality and the procedure's energy

consumption. Figure 1.1 presents the general classification of cutting procedures:

Compared to mechanical cutting, thermal cutting presents several advantages [11]:

• higher productivity;

• possibility to cut a very complex contour;

• possibility to mechanise and automate the process;

• reduced consumption of materials and energy;

• lower price.

There exist however also some disadvantages of thermal cutting [11]:

• it produces structural changes and changes in the chemical composition near

the cutting area;

• it produces remnant strains and stresses;

• the cut surface requires sometimes supplementary mechanical processing.

Around 85% of all thermal cutting operations are currently realised with gas flame

and oxygen [8, 87].

Due to the numerous technical-economic advantages, currently there is however

a tendency of replacing the cutting by burning with cutting by melting, especially with

plasma cutting or laser cutting.

8

DebitarePrin topire

Prin ardere

Cu daltă

Cu fierestrăumecanic

Cu foarfece

Prin aşchiere

Oxigaz

Oxiarc

Cu arc electric

Cu jet de plasmă

Cu laser

Cu fascicol deelectroni

Cu jet de apă

Cu jet abraziv

Electrochimică

Prin electroeroziune

Mecanică

Termică

Prin alteprocedee

Fig. 1.1: Classification of cutting procedures

Laser cutting started to be applied in the 70's, at first in speciality domains such

as aeronautics or military technics, but in the last couple of decades, due to the

numerous advantages, it gained a much broader range of applications.

The processing by laser beam displays a productivity that is 10...20 times

higher than that of the oxy-gas cutting or mechanical cutting procedures. Other major

advantages are [26, 29, 41, 47, 59, 79, 94, 144]:

• high cutting speeds;

• minimal material losses due to the small kerf (0.2...0.5 mm);

• high cutting precision;

• small heat-affected zone;

• large array of machinable materials.

9

Among the disadvantages of using a laser cutting system, there can be mentioned:

• higher capital costs than for other cutting systems;

• the maximal cuttable material thickness is 12 mm;

• cutting shiny metals such as aluminium or copper, where a large part of the laser

energy may be reflected outside of the cutting area, can be difficult.

Based on the technical, technological and economic characteristics of all cutting

systems presented, a comparative critical analysis was done. The main factors that

have to be taken into account, according to the speciality literature [98, 116, 120, 141,

154, 163] are:

• achievable cutting speed;

• edge precision;

• required precision level;

• types and material thicknesses that can be cut;

• dimension of the heat-affected zone;

• productivity;

• need for applying further machining stages on the cut surfaces;

• capital investment required for equipment, auxiliary materials etc.;

• operating costs;

• required skill level of the operators;

Thus, from the point of view of the cutting speed, the speciality studies indicate,

for all processed materials, a higher cutting speed with a laser beam, at lower material

thicknesses. At higher material thicknesses, however, the highest cutting speeds are

registered with plasma jet cutting. Figure 1.2 (adapted from [141]) presents the

evolution of cutting speeds for a mild steel, a stainless steel and aluminium, machined

with a laser beam of 4000 W, a plasma jet with a current intensity of 200 A and a

water jet of 50 HP, respectively. It should be added that plasma cutting systems

sometimes cut thinner metal sheets at lower speeds due to the choice of a smaller

current intensity which would lead to a better quality and precision of the cut.

Figure 1.3 (adapted from [141]) presents a comparative graph of the cutting

precision for the case of cutting with a laser beam of 4000 W, a plasma jet with a

current intensity of 200 A and a water jet of 50 HP, respectively.

It can be noticed that, regardless of the part thickness to be cut, the highest

precision is reached with laser cutting, while the large amount of heat set free during

plasma cutting leads to lower precisions.

10

a)

b)

c)

Fig. 1.2: Cutting speed for a part made of mild steel (a), stainless steel (b) and aluminium (c),function

of the material's thickness, for the case of cutting with a 4000 W laser, a 200 A plasma jet and a 50

HP waterjet, respectively.

Fig. 1.3: Cutting precision function of the material's thickness, for the case of cutting with a

4000 W laser, a 200 A plasma jet and a 50 HP waterjet, respectively.

With regard to the specific costs, table 1.1 [141] presents a comparative

overview, considering a cutting machine with a workspace of 1.8 x 3.6 m.

Table 1.1 Comparison of capital and operating costs

Cost element 4000 W laser

200 A high density

plasma burner

Water jet / Abrasive jet

with 50 HP pump

Capital costs 500.000 USD 115.000 USD 170.000 USD

Operating costs

10 USD/hour 10 USD/hour 10 USD/hour

Supplementary costs

2 - 15 USD/hour for auxiliary gas

3 USD/hour for auxiliary gas

15 USD/hour for the abrasive

11

As a preliminary conclusion, it can be said that, for the cutting of relatively thin

metal sheets, the best technology from the point of view of precision, cutting speed

and surface quality is laser cutting, but its main disadvantage is the high cost of the

equipment.

1.2 Structural analysis of the NC cutting machines

The goal of a machine tool is to process parts by cutting, under certain

conditions of productivity, dimensional precision and surface quality. If the processing

orders are programmed with a numerical control equipment, the machine is called a

numerical control machine tool (NCMT).

Regardless of the cutting procedure employed, the components of a cutting machine

must provide following functions [30, 92, 138]:

• bringing and fastening the blanks in the work area;

• moving the work units with precisions as high as possible;

• providing an optimal distance between work unit and blank;

• tuning and maintaining the cutting parameters at optimal levels;

• cutting the blanks under optimal quality and productivity conditions.

The cutting machines, or more precisely the numerical control equipment

controlling them, must allow movements along three axes: x, y, z. Each of these

movements is done by means of a kinematic chain. The generalised functional scheme

of a feed kinematic chain used in the structure of cutting machines is presented in

figure 1.4.

Fig. 1.4: Generalised functional scheme of a feed kinematic chain

M - direct current electrical motor; MR – control mechanism;

Sig – safety mechanism; RT – trajectory changing mechanism;

PO – start/stop mechanism; UL - work unit.

I – inversion mechanism;

12

The design of an actuation system for such feed kinematic chains in the case of

modern cutting machines, where the tools is materialised by an energy beam, must

take into account the technological particularities of such machines:

• Minimal resistive forces and moments;

• Work strokes and feed rates that can vary in a relatively broad range.

Therefore, the feed system has to provide both an adequate functioning from

the point of view of stability and safety, as well as an adequate productivity. Moreover,

a high programming flexibility must be provided, in order to be able to change quickly

and efficiently the parameters of the work cycle.

1.3 Motion control systems used for cutting machines. Numerical axes

The notion of motion control defines the control of the movement's kinematic

parameters (position, speed, acceleration), as well as their correlation for the case of

motions controlled on several directions.

Generally, the structure of a motion control system has to include following

subsystems (figure 1.5):

Fig. 1.5: The block diagram of a motion control system

• control block;

• amplifier block;

• actuation block;

• mechanical motion support;

• reaction transducers;

• interface elements.

Numerical axes must provide following functions [118]:

13

displacing the work unit with the imposed feed rate, in conditions of

achieving a good regularity in time and space;

control of feed rates for a broad range of these (from a technological feed rate

in the order of mm/min, for very precise applications, to tens of m/min, for the

case of rapid positioning of mobile elements of NCMTs);

tracking the feed rate variation in the shortest possible times;

precise control of the positions of work units on each axis and precise coordination

of several axes in the case of contouring operations;

providing moments capable of surpassing all resistance forces that occur in

the process; any variation of this moment must not influence the positioning

or contouring precision and must not lead to variations in the feed rate.

The block diagram of a secondary loop for the closed-loop control of the feed

rate is presented in figure 1.6.

Fig. 1.6: Block diagram for the closed-loop control of the feed rate

In this case, a direct current servo-motor is used. The power amplifier generates the

voltage level Ub by amplifying the feed rate error signal, this being the input parameter for

the current loop. Most often, the current loop may be assimilated with a proportional

block, with a proportionality constant I/R, where R is the resistivity of the DC motor's

armature. The servomotor is controlled by means of the current through the motor Im.

Under the action of this variable, the motor produces the couple needed for accelerating

the moving masses until reaching the imposed angular speed ω. The motor couple thus

generated cancels the effect of disturbances in the system (occurring as resistive moments

and friction moments). Therefore, the usage of DC servomotors represents an optimal

solution for the actuation of this type of systems.

14

1.4 Aspects of the analysis of automated systems from NC machines

If a nonlinear function yi is considered, representing the dynamic behaviour of the element i from the system,

)( ii xfy = , (1.1)

the differential equation of the global system can be written as:

0)(,...,)(

,)(

),(,...,)(

,)(

1

1

1

1

=

−

−

−

−

txdt

txd

dt

txdty

dt

tyd

dt

tydf

m

m

m

m

n

n

n

n

, (1.2)

or:

x(t)b + dt

dx(t)b+ ... +

dt

x(t)db

+ dt

x(t)db =y(t)a +

dt

dy(t)a ... +

dt

y(t)da +

dt

y(t)da

01-1m

-1m

-1m

m

m

m01-1n

-1n

-1nn

n

n +

. (1.3)

The general solution of this equation is of the form: )()()()( tytytYty sftftl ++= , (1.4)

where: ytl(t) is the free transitory component, which does not depend on the input, being determined only by the system's dynamics and by the initial non-zero output conditions;

ytf(t) is the forced transitory component, depending both on the system's

dynamics and on the input;

ysf(t) is the forced component in stabilised regime.

Differential equations such as the one described by (1.3) can be solved relatively

easily using the Laplace transform. Then, the differential equation is changed into a

polynomial equation:

If: ., Y(s)s=F(s) atunci dt

y(t)d=f(t) n

n

n

(1.5)

In the current case, by applying the Laplace transform to the initial differential equation, there results:

.X(s)b+ sX(s)b +...+ X(s)sb +

X(s)sb =Y(s)a + sY(s)a +...+ Y(s)sa + Y(s)sa

01-1m

-1m

mm01

-1n-1n

nn +

(1.6)

The transfer function can be defined as ratio between the Laplace transform of the output parameter and the Laplace transform of the input parameter, for initial zero conditions:

a+sa+...+sa+sa

b+sb+...+sb+sb=

X(s)

Y(s)=H(s)

011-n

1-nn

n

011-m

1-mm

m . (1.7)

15

1.5 Study of the dynamic behaviour and of the kinematic precision in

NC cutting machines

Any error occurring in the functioning of a motion control system is

automatically transformed into errors when generating the parts' contour. Experimental

researches presented in the speciality literature [2, 9, 16, 21, 118, 127] indicated that

errors introduced by the functioning of numerical axes are the main component in the

total amount of errors occurring during the functioning of a NC machine-tool.

Among the factors significantly influencing the dynamic behaviour and the

kinematic precision of numerical axes an important role is held by the dynamic constraints.

Within this category of factors there needs to be mentioned firstly the system's dynamic

untuning. It can manifest itself through the existence of different amplifying factors on each

axis, case in which especially the contouring error in stationary regime is affected, or by

different time constants on each axis, case in which the transitory regime is negatively

affected. An element that would require a more in-depth study, as yet not fully solved

[118, 127], is the presence or absence of contouring errors in the case of perfectly

tuned numerical axes. Such errors occur for example when a circular contour is cut, it

being deformed as an ellipse along the axis with a smaller amplifying factor.

When cutting parts with 90º corners at constant speed (without stopping in the

corner point), contouring errors are noticeable that depends on the dynamic behaviour of

the numerical axes.

The constructive and functional elements that allow the optimisation of the

dynamic behaviour from the point of view of work parameters stability are the

controllers. These are feedback control systems with the role of maintaining a

constant input value during the functioning cycle of the controlled system.

A PID controller is characterised by the fact that the correction signal at its output

represents a combination of three components: a proportional component (P), an integral

one (I) and a derivative one (D), resulting from the processing of the error signal [42].

The functioning of a feed-forward controller is relatively simple: a transfer

function G0-1

(z) is introduced in the control system, that is the precise reverse function of

the system's global transfer function G(z), so that:

G0-1

(z) · G(z) = 1, (1.8)

and the reference input value (the imposed value) becomes equal to the system's output

value (the realised value) [8, 58, 118]. Feed-forward controllers reduce mainly the

tracking errors Ex and Ey on the two axes, but the contouring error ε is indirectly

reduced too as a result [118].

16

1.6 Preliminary conclusions and objectives of the thesis

As a consequence of the facts presented above, the current thesis aims at carrying

out theoretical and experimental researches regarding the evaluation of dynamic behaviour

and kinematic precision of feed kinematic chains from the structure of NC laser cutting

equipments, as well as the investigation of means to improve these elements.

The thesis' proposed objectives are:

From the point of view of theoretical researches

• elaborating a mathematical model for the numerical axes using as execution

element the DC servomotor. As has been shown earlier, this constructive solution is

currently the best technical-economic compromise for this kind of equipment;

• taking into account at the model's construction of the influence of as many

disturbance factors as possible (masses, inertia forces, possible technological

forces). There is an obvious need for the modelling results to be as close as possible

to the studied system's real behaviour. For this, it is necessary for the modelling to

take into account as many as possible of the influences exerted during functioning

on the numerical axes;

• introducing in the model's structure of the influence of numerical control (discrete

transfer functions, sampling periods, numerical controllers). The methods being

currently applied in studying linear continuous systems cannot be used also for

studying numerical axes. Computer control implies not only a structural modification

of servosystems for the automated position control, but it is also necessary to tackle

this study with methods specific for automated control systems with sampling;

• elaborating mathematical models for generating reference parameters, both in

rapid positioning regime and in linear and circular interpolation regime. The

functioning time of numerical axes, in either of the two regimes, is

approximately equal, so both situations have to be studied. Furthermore,

kinematic and dynamic conditions differ significantly in the two cases, requiring

their separate tackling;

• study through simulation of the dynamic behaviour and of the kinematic precision of

numerical axes. The study through simulation aims at emphasising the influence of

various factors (constructive solutions employed, variation of the kinematic

parameters, disturbance factors etc.) on the numerical axes' behaviour. Also, though

simulation there will be identified the main methods and techniques for improving

the axes' behaviour.

17

From the point of view of experimental researches

• studying the influence of the various categories of technological factors,

including the control level at the numerical control, on the structural,

mechanical and surface quality characteristics of parts processed by cutting.

There will be realised by cutting with energy beams (oxy-gas, laser), from

sheet-type blanks, with different cutting regimes, parts that will later be

analysed not only from a dimensional and shape point of view, but also from the

point of view of microstructure, microhardness and roughness.

• experimental research of the precision of numerical axes from the structure of

NC machines for cutting with energy beams, in positioning regime and in

contouring regime. This implies the actual unfolding of determinations,

following an experimental programme according to the current norms, including

the processing of the acquired data;

• experimental validation of the theoretical methods for improving the dynamic

behaviour and the kinematic precision of numerical axes. The validity of solutions

resulting from theoretical researches will be checked, by carrying out the needed

modifications on the experimental system, in order to improve its behaviour.

18

Chapter 2

THE MATHEMATICAL MODEL OF A FEED DRIVE FROM

THE STRUCTURE OF NUMERICAL-CONTROL CUTTING

MACHINES

2.1 General considerations regarding the mathematical modelling

The behaviour of an automated system, under given exploitation conditions, may

be described by group of algebraic and differential equations that express the dependence

of the output parameter on the input parameter. Based on this, a mathematical model can

be obtained, that allows not only the description of the system's current behaviour, but

also the behaviour in conditions of modified working conditions.

In order to define a mathematical model for a numerical axis, it is necessary to

go through following steps [118]:

1. Decomposing the analysed system in its basic components that take part in realising its functionality and equating these components with pure (and if possible linear) dynamic elements;

2. Writing the characteristic equations for each of these elements; 3. Linearising these equations (in order to achieve a linear global system); 4. Determining the interconnection relationships between the component dynamic

elements. By eliminating the intermediate variables, one can determined the differential

equation of the global system. This equation (defined through relation (1.2) in the previous chapter), together with the initial conditions: values of x and y parameters and of their derivatives function of time at the moment t = t0, composes the system's mathematical model.

2.2. Structure of the studied numerical axis

Starting from the basic elements presented in the previous chapter and from the

idea that an actuation based on DC motors is more efficient than the one based on AC

motors, the theoretical researches regarding the modelling and simulation of a

numerical axis' control, presented in the following, were based on the actual case of a

feed kinematic chain having as execution element a DC servomotor.

19

The principle structure of such a numerical axis is presented in figure 2.1. In

this case, there was adopted the solution of mounting the incremental transducer on the

servomotor's shaft.

Fig. 2.1: Structure of a numerical axis with DC servomotor

RP - position controller; I – integrator;

CNA - numerical-analogical converter; TG - speed voltage generator;

A – amplifier; TID - incremental displacement transducer

M - DC servomotor

The position reference θc, which in the context of the systemic approach

represents the main input parameter in the system constituted by the numerical axis, is

generated at time intervals equal to the sampling period T by the control equipment's

interpolator. It is then compared to the feedback parameter from the incremental

transducer, sampled at the same time interval.

The output parameter in this system is the effective displacement of the work unit on

the considered direction.

2.3. Determining of the relationships that characterise the functioning

of the numerical axis

In order to correctly determine the analysed numerical axis' characteristics, the

starting point has to be the axis' global transfer function. For determining this function,

there need to be taken into account, according to the theory of automated control, the

transfer functions of each component, taken separately.

20

For a DC motor, the system's transfer function for the disturbance (static

moment Ms) is:

sτ+

KKRB+

R

= (s)M

ω(s) =

)ML(

L(ω

m

vt

ss ⋅1)

. (2.1)

Following the calculations, in the closed speed control system, the revolution

speed can be written as:

][ (s)MKK+RB

R-

KK+RB

KKU(s)

s+1=(s) s

vtvt

ta

⋅τ

αω (2.2)

or

s+1

(s)MK-U(s)K=(s)

s21

τω . (2.3)

where two components of the transfer function of the closed speed loop can be noticed:

function of the input (control voltage) and function of the disturbance (static moment).

Applying the Laplace transform, the motor's revolution speed ω(t) could be

written as:

)e-(1)MKK+RB

RK-U

KK+RB

KK(=(t)

t-

s

vt

t

vt

ta τ⋅α⋅αω . (2.4)

At maximal values of the control voltage Um and of the static moment Msm, the

motor's revolution speed must reach a maximal value ωm:

MKK+RB

RK-U

KK+RB

KK= sm

vt

tm

vt

tam

⋅α⋅αω . (2.5)

The block diagram of the numerical axis is presented in figure 2.2.

Fig. 2.2: Block diagram of the numerical axis

21

The incremental transducer is characterised by its amplifying coefficient Ke,

defined by the number of impulses sent during a full rotation of the lead screw:

[imp/rad] 2

N=K

imp

eπ

, (2.6)

where Nimp is the number of impulses emitted by the transducer during a rotation.

The Laplace transform of the position error can be expressed as:

K+s+s

(s)MKK+(s)x)ss+(1K

K

=E(s)2

sg2r

g

e

⋅τ

τ ][

. (2.7)

Relation (2.7) allows the distinguishing of the two components of the transfer

function, relative to the input and relative to the disturbance:

K+s+s

s)+s(K

K

= (s)x

E(s) = (s)H 2

2

g

e

r

0⋅τ

τ

; (2.8)

K+s+s

KK =

(s)M

E(s) = (s)H 2

2e

s

0Pτ

. (2.9)

These two relations represent the global transfer functions of the numerical axis

with DC servomotor and will be used subsequently for its mathematical modelling.

2.4. Analysis of the kinematic precision of the numerical axis

The problem of motion control of numerical axes in the case of cutting

machines must be discussed for two distinct situations:

- the case of rapid positioning motions (point by point positioning); during such

displacements there are no machining operations, reaching of the final point under the most

accurate conditions in the shortest possible time being most important;

- the case of machining (contouring) motions, carried out with the technological

feed rate, on one or several axes; in this case, the realised trajectory and the accuracy with

which it reproduces the imposed trajectory, is essential.

An important parameter in this regard is the tracking error, defined as distance

between the instant values of the realised position and of the imposed position.

Figure 2.6 presents the variation of the kinematic parameters (distance covered,

speed/feed rate, acceleration) for a point by point positioning cycle on a movement axis.

22

Fig. 2.3: Distance covered, speed/feed rate, acceleration) for a point by point positioning cycle

Tac, sac - acceleration time, distance covered during acceleration

Tvc, svc - duration of movement at constant speed, distance covered with constant speed;

Tdc, scc - duration of deceleration, distance covered during deceleration;

For a positioning displacement, it is easy to realise in practice a trapezoidal

speed profile, but it presents the disadvantage that the acceleration varies abruptly,

which might pose problems especially during contouring operations, through the

occurrence of the "jerking" phenomenon, which influences negatively the accuracy of

the realised contour and introduces vibrations into the system.

Nevertheless, if considering displacements on only one axis, the trapezoidal

profile is the most used one, being easier to generate than a paraboloidal one and

therefore it is found with all motion-controller-type industrial system. Moreover,

in practice the jerking is considerably diminished, because the delaying effects

introduced by various elements of the control system determine the actual shape

to be close to the paraboloid one.

Considering the acceleration equal in absolute value on the acceleration and

deceleration phases, respectively, and considering also the periods ta and td equal, between

the kinematic parameters for a trapezoidal profile there can be written following relations:

23

a

vtt da

max== ; (2.10)

)(max

dact

ct ttTv

st +−== ; (2.11)

22

maxmax dada

tvtvss

⋅=

⋅== ; (2.12)

a

vss da

⋅==

2max ; (2.13)

ctct tvs ⋅= max , (2.14)

where: a is the acceleration during the acceleration and deceleration phases;

sa – distance covered during the acceleration period;

sd – distance covered during the deceleration period;

sct – distance covered during the period of movement at constant speed.

The contouring regime implies the correlation in time of the displacements on

the numerically controlled machine axes and involves the introduction in the

equipment's memory and the processing of a very large amount of coordinates of

points from the trajectory that defines the constructive shape of the part.

For practical reasons, however, the trajectory is decomposed in elementary curves, and in the equipment's memory there are introduced only the coordinates of the initial and final points, the coordinates of intermediate points being calculated by the equipment by means of interpolation algorithms. These can be easily implemented in the software. Currently there are mainly two types of interpolators:

• interpolators using the reference impulses technique;

• interpolators using the binary words technique. The reference impulses technique, the first to be implemented, was

implemented by means of two method: the method of differential digital analysis (DDA) and the method of direct function calculus (DFC). These methods have however the disadvantage of using a large amount of mathematical operations (in the case of the DDA method) or of logical operations (in the case of the DFC method). Therefore currently there is used especially the binary words (reference words) technique, which will be also tackled in more detail in the following. In the case of circular interpolation, the values of reference words for the control of the two axes, at the moment Tk, can be described with the relations:

⋅π

=

⋅π

=

,2

cos

;2

sin

kk

kk

TT

Ry

TT

Rx

(2.15)

24

where: R is the radius of the circle of which the arc is a part; T – period of the sine and cosine functions (depending on the feed rate on the axis). Tk can be determined function of the sampling period according to the relation:

sk TkT ⋅= . (2.16)

The angular displacement increment θincr is determined with the relation:

ssincr TR

vT

T⋅=⋅

π=θ

2, (2.17)

and the linear displacement increment lincr with the relation:

sincr

incr Tv

Rlπ

=π

θ⋅=

22 , (2.18)

The equations found within this chapter for realising the mathematical model were used in the next stage of theoretical researches, for the numerical simulation of the behaviour of a feed system from the structure of a laser cutting machine, presented in chapter 3 of this thesis, for the calculus of the optimal functioning parameters of the DC servomotor used for actuation.

Thus, the equations characterising the functioning of the numerical axis were implemented for the actual case of the system with DC servomotor.

Also, the mathematical model that was determined here, was used also for the configuration and functional optimisation of the system employed for experimental researches, detailed in chapter 4 of this thesis.

25

Chapter 3

STUDY BY SIMULATION OF THE BEHAVIOUR OF

THE FEED SYSTEM FROM THE STRUCTURE OF A NC

LASER CUTTING MACHINE

3.1 Generalities regarding the analysis through simulation of the

numerical axes

Numerical simulation is a work method in systems analysis, by which, with the

help of adequate hardware and software, a system's model can be evaluated over a period of

time, collecting data on the setup and functioning of the model under various conditions and

using those data for estimating the real characteristics of the analysed system [75].

The simulations that were carried and that are presented in the following were

realised with the help of the MATLAB & Simulink software package, version 7.0.

MATLAB (MATrix LABoratory) is a performant, interactive software

environment, destined for engineering and scientific numerical calculations, that

appeared in a first shape at the end of the 70's and has been extended considerably in

the last few year. Its declared goal is to help the users to solve a large array of

analytical and numerical problems using methods based on matrix calculations,

offering easy access and simple implementation possibilities for state-of-the-art

numerical algorithms [82, 148, 161].

Simulink is a software for modelling, simulating and analysing linear or nonlinear,

continuous, discrete or hybrid dynamic systems, integrated with the Matlab software.

3.2. Structure and characteristic parameters of the studied numerical axis

In order to reflect the influence of the numerical control on the modelled system

and the model's hybrid character, the functional block diagram used for the study by

simulation of the behaviour of a numerical axis with DC servomotor, realised in the

Matlab & Simulink software package was completed with following elements (fig. 3.1):

- an extrapolation block (zero-order extrapolator) on the direct way;

- a sampling block on the feedback loop. for sampling the quantified signal

from the transducer.

26

v

Viteza

t

Timp

Ms

Uc

Omega

Subsistem

bucla de viteza

Kp

Regulator

pozitieKg

Kg x

Pozitia

Moment rezistent

static1

s

Integrator

pozitieKe1/Kg

Extrapolator

de ordinul zero

Extrapolare si retinere

semnal de reactie

Ke

Encoder

du/dt

Derivative

Kc

Convertor

numeric/analogic

Comparator

pozitie

Baza de timp

Afisaj viteza

Afisaj pozitie

Afisaj acceleratie

a

Acceleratia

1

Fig. 3.1: Block diagram used for the study by simulation of the numerical axis' behaviour

It was decided to analyse the numerical axis' dynamic behaviour both in

positioning regime and in contouring regime, by simulation.

Therefore, in the following there have been determined the values of parameters

needed for simulating the functioning of a numerical axis using a DC servomotor of

type Sanyo T 730-012 [158].

The mentioned servomotor has following characteristics:

- nominal couple Mn = 1,18 Nm;

- revolution speed nn = 2500 rpm;

- nominal current intensity In = 5,2 A;

- maximal impulse current intensity Imax = 40 A;

- nominal voltage Un = 75 V;

- resistivity at the terminals la borne Ra = 1,1 Ω;

- constant of the motor couple Kt = 0,273 Nm/A;

- speed coefficient Kv = 28,6 . 10-3 V/min-1;

- inertia moment Jm = 0,270 . 10-3 kgm2;

- viscous friction coefficient Bm = 0,039 . 10-3 Nm/min-1.

The speed voltage generator constant is: Kth = 7 . 10-3 V/min-1.

27

In the following the inertia moment for the whole system, reduced to the engine

shaft was calculated, considering that the maximal allowable speed on the axis is

vmax= 5 m/min = 0.0833 m/s:

001165,000027,000089532,000000088,0 =++=++= mspt JJJJ kgm2.

Similarly, there were determined the values of the maximal angular acceleration that

could be developed by the engine, of the linear acceleration, of the loop's time constant,

attenuation factor, amplifying factor and damping factor.

Based on these calculations it was possible to generate a data set used for

simulating the functioning of the numerical axis.

3.3 Study by simulation of the dynamic behaviour in positioning regime

In this research stage it was sought to analyse the behaviour of the numerical

axis with DC servomotor in positioning regime, at the displacement on a single

movement direction, in fast approach-retreat regime (in the absence of technological

forces) and in technological feed regime (in the presence of technological forces).

The necessary reference kinematic parameters (distance, speed (feed rate),

acceleration), were generated with the help of a Matlab program, at time intervals of 1

ms (10 × sampling period), following a trapezoidal speed profile.

Figure 3.2 shows the variations of the reference kinematic parameters for a

rapid approach movement and for a technological feed movement, respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6x 10

-3

Pozitie

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.01

0.02

Viteza

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.5

0

0.5

Accele

ratie

Timp (s)

Fig. 3.2: Reference kinematic parameters for a technological feed movement

28

Using the results obtained following the mathematical calculations carried

out in paragraph 3.1, in the following there was simulated the system's dynamic

behaviour both for a fast approach and for a technological displacement, with the

help of the Matlab v.7.0 software and using the diagrams developed in Simulink

and presented in chapter 2 of the current doctoral thesis.

The results of simulations (the graphical display of the variation of output

parameters function of time) are presented in figures 3.3 - 3.5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

Timp [s]

Spatiu [

m]

Spatiul parcurs pe cursa rapida

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Viteza pe cursa rapida

Timp [s]

Viteza [

m/s

]

Fig. 3.3: Distance covered during the rapid

approach

Fig. 3.4: Speed during the rapid approach

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Timp [s]

Accele

ratia [

m/s

2]

Acceleratia pe cursa rapida

Fig.3.5: Acceleration during the rapid approach

As can be seen especially from the speed and acceleration graphs above, the system's dynamic behaviour during the technological feed regime is unsatisfactory, because there exist very large oscillations.

An efficient method for tackling this problem, recommended also by the speciality literature [27, 33, 118, 139], is the tuning of the controller as for the case of a

29

continuous system. In this respect, the loop's amplifying factor was considered constant, both for the case of a continuous system and for the case of the hybrid system.

The amplifying factor is determined with the relation:

en

p

pd

KU

KK

⋅

=

−12

. (3.1)

where: Kpd is the controller's amplifying factor in the digital system variant; Kp - the controller's amplifying factor in the continuous system variant; KBV - the speed loop's amplifying factor.

Replacing with the corresponding numerical values, the amplifying factor is:

39,386506725000009765,0

15846,96

25002

815846,96

114

=⋅

=

⋅

=

−

pdK

The simulation was then redone with this new value of the amplifying factor. It can be noticed that under the new conditions, the oscillations are much smaller and there were no positioning errors.

In the following, with the tuned system, its behaviour during a technological feed was tested in the presence of a resistive moment of 1 Nm, as disturbance factor in the analysed system. It can be noticed that the disturbance is being annulled quickly.

The next step comprised the simulation of the system's dynamic behaviour, starting from the mathematical model that was presented earlier. For this, in a first phase there were analysed the system's reactions to a position ramp-type signal (as well as speed step and acceleration impulse), both in the presence and in the absence of a static resistive moment. Based on these results, following preliminary conclusions regarding the behaviour

of numerical axes with DC servomotor within NC cutting systems can be drawn:

• The tuning of controllers is a complicated process that has to be carried out

both analytically and by successive trials;

• The simulated behaviour in the two functioning regimes is in agreement with the

requirements for numerical axes: relatively small and constant, thus easily

compensable positioning errors (in the order of tens of micrometers), good

dynamic behaviour;

• The system is very robust with regard to resistive-moment-type disturbances

(positioning and speed errors are basically constant for resistive moments

between 0 – 1 Nm).

Based on the realised model and simulation diagram and on the above conclusions,

it was possible to tackle the next steps of researches, which imply the study by simulation of

the multiaxial movement and the unfolding of corresponding experimental researches.

30

3.4 Study by simulation of the generation, by cutting, of 90°°°° corners

The generation of 900 corners, without stopping in the corner point, marks one

of the situations in which the numerical axes' dynamic behaviour decisively influences

the machine's contouring precision.

The quality of the transitory regime (presence or absence of overregulation,

size of increasing and of stabilisation times etc.) influences the errors in the corner

point. It must be determined, however, whether the error at corner generation is

reduced only to the error in the corner point, determined by the behaviour of the

numerical axes in transitory regime, or whether there exists also an error determined

by the behaviour of the two orthogonal numerical axes in stationary regime. For

studying this situation through simulation, there has been realised in Simulink the

diagram presented in figure 3.6.

vy

Viteza axa y

vx

Viteza axa x

t

Timp

Subsistemul

axei y

Subsistemul

axei x

Out

Sistem abilitat

pentru generarea

impulsurilor pe axa Y

Out

Sistem abilitat

pentru generarea

impulsurilor pe axa X

Referinta

y

Pozitia axa y

x

Pozitia axa x

>

Operator

relational

axa Y

<=

Operator

relational

axa X

refy

Cuvinte de referinta

axa Y

refx


axa X

11

12:34

Ceas digital

Actuala

ay

Acceleratia axa y

ax

Acceleratia axa x

Fig. 3.6: Block diagram for simulating the generation of 90

0 corners

31

In the following, the generation of corners with an identical amplifying factor of the position controller on both orthogonal axes: Kpdx=Kpdy= 39.38 was studied, considering at first that the resistant moment is 0. This leads to the graphs of displacement, speed and acceleration shown in figures 3.7 – 3.11. The system obviously had an unsatisfactory behaviour in the corner point, so the profile was also negatively influenced.

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02

0

5

10

15

20

x 10-4

Timp [s]

Spatiul [m

]

Spatiul parcurs pe X

Ep = 2.86*10-4

distanta impusa

distanta parcursa

__

__

0.98 0.99 1 1.01 1.02 1.03 1.04

0.093

0.094

0.095

0.096

0.097

0.098

0.099

0.1

0.101

Timp [s]

Spatiu [

m]

Spatiul parcurs pe Y

Ep = 1.32*10-4

distanta impusa

distanta parcursa__

__

Fig. 3.7:Distance covered on the x axis Fig. 3.8: Distance covered on the y axis

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Timp [s]

Viteza [

m/s

]

Variatia vitezei in functie de timp

viteaza pe x

viteza pe y__

__

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-25

-20

-15

-10

-5

0

5

10

15

20

25

Timp [s]

Accele

ratia [

m/s

2]

Variatia acceleratiei in functie de timp

acceleratia pe x

acceleratia pe y__

__

Fig. 3.9: Speed variation function of time Fig. 3.10: Acceleration variation function of time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-15

-10

-5

0

5

10

15

Timp [s]

Cure

ntu

l [A

]

Variatia curentului in functie de timp

curentul pe x

curentul pe y__

__

Fig. 3.11: Current intensity variation function of time

32

Next, a resistant moment due to inertia forces was taken into account, with the values on the x and y axis: Mix= 0.23 Nm and Miy= 0.69 Nm. Following the simulation it was noticed that the error Ex increases, reaching a value of 0.0005 m and also an error appeared on the y axis. While being small, Ey = 0.0001 m, it represents a shape error and thus influences the part's dimensional precision.

First it was attempted to reduce the system's oscillations by introducing in the position controller a derivative component, both for the x axis and for the y axis. The simulations run with this setup proved however that this derivative component did not influence the contour errors, they maintaining the initial values.

Another solution studied for improving the system's behaviour was the cutting with different amplifying factors on the two axes. In order to study this possibility, the damping factor on the x axis was kept at the value 1=ξ , while on the y axis, a damping factor of

2,1=ξ was implemented. It was noticed that the positioning errors decrease both on the x

axis, Ex = 5.48 .10-4 m and on the y axis, Ey=1.06 .10-4 m, the acceleration on the y axis decreases to aymax= 9.78 m/s2, and the current intensity is also smaller, remaining within the limits imposed by the employed motor. Most importantly, the speed's oscillation is very small, of about 0.1 m/s. The actual results are presented in figures 3.12 - 3.14.

0 1 2 3 4 5 6 7 8 9 10

x 10-3

0.091

0.092

0.093

0.094

0.095

0.096

0.097

0.098

0.099

0.1

X (BLU=0.001)

Y (

BLU

=0.0

01)

Generarea unui colt (Kpdx=19.69; Kpdy=13.68)

traiectorie programata

traiectorie realizata__

__

Fig. 3.12: Cutting of the 90

0 corner with Kpdx = 19.69; Kpdy=13.68

0 1 2 3 4 5 6 7 8

x 10-4

0.099

0.0991

0.0992

0.0993

0.0994

0.0995

0.0996

0.0997

0.0998

0.0999

0.1

X (BLU=0.001)

Y (

BLU

=0.0

01)




__

0.0592 0.0594 0.0596 0.0598 0.06 0.0602

0.099

0.0992

0.0994

0.0996

0.0998

0.1

0.1002

X (BLU=0.001)

Y (

BLU

=0.0

01)



traiectorie realizata

__

__

Ex

Ey

Fig. 3.13: Detail from fig.3.12 (corner area) Fig. 3.14: Detail from fig. 3.12 (error)

33

3.5 Study by simulation of the generation, by cutting, of circular contours

This chapter presents studies with regard to the numerical simulation, with the

Matlab & Simulink software package, of the generation of a circle with the radius of

30 mm, whereby it was attempted to minimise the contouring error by modifying

the technological parameters and the parameters of the motion control system.

In a first stage, there was studied the evolution of the contouring error for a feed rate

along the axis of 0.025 m/s and a sampling period of 0.001 s. Then the feed rate was

modified in steps to the value of 0.0133 m/s and the simulations were redone in order to

determine to what extent the feed rate influences the contouring error.

Then, as for the generation of the 90° corner, it was attempted to optimise the system

from the point of view of a reduced contouring error, by modifying the amplifying factor on

one axis at the time from Kpd = 39.38 la Kpd = 19.69. Also, a random disturbance was

introduced, which varies between a minimal value and a maximal value equal to the

laser beam radius of 0.02 m.

vy

Viteza axa y

vx

Viteza axa x

t

Timp

Subsistemul

axei x1

Subsistemul

axei x

Referinta

y

Pozitia axa y

x

Pozitia axa x

Generare

cuvinte de referinta

axa x

Generare

cuvinte de referinta

axa y

11

refy


axa Y

refx


axa X

0

30

12:34

Ceas digital

Actuala

ay

Acceleratia axa y

ax

Acceleratia axa x

Fig. 3.15: Block diagram used for simulating the generation through cutting of a circle

34

The simulation was based on the block diagram presented in figure 3.15,

realised with the Simulink software.

Figures 3.16 and 3.17 present the trajectory resulting when using a feed rate of

0.025 m/s, comparatively with the programmed trajectory and a detail of those trajectories.

As can be seen in figure 3.16, there exists a contouring error between the generated profile

and the programmed one, but it is small enough to not be visible for the scale used in fig.

3.17. The maximal value of the contouring error on the x axis is of 0.0672 mm, while the

maximal error on the y axis is of 0.3449 mm.

0 10 20 30 40 50 60 70

-30

-20

-10

0

10

20

30

Axa X

Axa Y

Eroarea de pozitie la generarea unei traiectorii circulare



__

58.8 59 59.2 59.4 59.6 59.8

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Axa X

Axa Y

Eroarea de pozitie la generarea unei traiectorii circulare



__

Fig. 3.16: Positioning error at the generation of

a circular trajectory for v = 0.025 m/s

Fig. 3.17: Detail from fig.3.16

An important factor to be analysed in this case is the influence of the feed rate

on the contouring error. For this, the feed rate has again been varied in steps. It could

be noticed that, by decreasing the feed rate from 0.025 m/s la 0.0133 m/s, the contouring

errors also decrease to approximately half the initial values, their maximal values being

Exmax = 0.0358 mm and Eymax = 0.1839 mm. It can be therefore said that between the feed

rate and contouring error there is an almost linear dependence.

However, decreasing the cutting speed as method of increasing the cutting precision

is suitable only for complex parts or for parts produced in small batches. In the case of large

batches, it is not accepted by producers, as it considerably reduces the machine's

productivity.

Therefore, another way to optimise the cutting conditions with regard to the

trajectory generation precision. So, while keeping the feed rate at 0.025 m/s, the

amplifying factor was reduced successively on one axis at the time, from 39.38 to

19.69.

35

When reducing the amplifying factor on the x axis, the contouring error on the x

axis decreases to a maximal value Exmax = 0.0336 mm, while on the y axis the error

remains unchanged Eymax = 0.344 mm. When modifying the amplifying factor on the y

axis, no improvement is noticed, to the contrary, the maximal error on the y axis

reaches 0.6898 mm.

As a conclusion of the simulations carried out and described in this subchapter,

it can be concluded that in the case of circular interpolation, if it is sought to maintain

a high productivity, the best solution is to adopt different amplifying factors on the two

axes.

36

Chapter 4

EXPERIMENTAL RESEARCHES REGARDING THE

OPTIMISATION OF THE DYNAMIC BEHAVIOUR OF

KINEMATIC FEED CHAINS FROM THE STRUCTURE OF

NUMERICAL CONTROL CUTTING MACHINES

The experimental researches carried out within this doctoral thesis have

targeted the practical determination of the optimal setup parameters at a NC cutting

machine and, at the same time, the validation of the corresponding results achieved

through simulation with the help of the software package Matlab & Simulink and

presented in chapter 3 of the thesis.

These researches were carried out on machines and equipments from the

Faculty of Engineering of the "Lucian Blaga" University of Sibiu and from the company

S.C. COMPA S.A. Sibiu, respectively.

4.1 Equipments used for the experimental researches

The technological system used for the unfolding of experimental

researches comprised a NC machine tool for laser cutting MAZAK NT-X48

Champion, an oxy-gas cutting machine OXYTOME HPC 20 and two coordinate

measuring machines ZEISS PRISMO 7 and ZEISS CONTOURECORD 1600 D,

respectively.

The cutting machine OXYTOME HPC 20, made by the company SAF-FRO,

component of the AirLiquide Group [151], is an automated portal-type machine that

can cut metal sheets with a width of up to 2425 mm. It can be equipped on demand

also with a Nertajet HP plasma cutting system. The control system HPC Digital Process

allows a continuous monitoring of the gas flow and of the cutting process [10], while the

cutters are equipped with pilot valves for preheating, leading to a higher productivity.

In order to carry out the experimental researches with regard to laser cutting, which

are the main part of the researches presented in the following in this chapter, a laser cutting

machine MAZAK NT-X48 Champion manufactured by Yamazaki Mazak Optonics

Europe [165] was used, from the endowment of the company S.C. COMPA S.A Sibiu. The

machine is equipped with a CO2 laser.

37

Given the fact that the numerical control unit Mazatrol L-32B of this machine was

defective and the company did not want to invest at that moment in a new such unit, it was

attempted to replace it with a numerical control unit Fagor 8055 from a NC milling machine.

This replacement was justified by the fact that the company DANOBAT from Spain

produces laser cutting machines equipped with Fagor 8055 NC units [166].

The numerical control unit FAGOR 8055 [142, 166] is realised in a modular

system and can control up to 7 numerical axes and 2 shafts. It allows block processing

times of up to 1,5 ms, can analyse the cutting head trajectory with up to 75 blocks in

advance and offers the possibility of an advanced control of the displacement for

avoiding the occurrence of "jerking". This unit uses DC servomotors for the control of

numerical axes.

Prismo 7 S-Acc is a portal-type coordinate measuring machine, manufactured

by the company Zeiss from Germany. It represents an improved version of the Prismo

Navigator measuring system, is equipped with a VAST measuring device and allows

the active scanning of the parts to be measured [156].

The CONTOURECORD 1600D measuring machine, also manufactured by

the company Zeiss from Germany, is designed for the measuring and analysis of the

contours of parts [143]. Similarly to the Calypso software from the Prismo S-Acc

machine, the TiMS software integrated on the Contourecord machine and functioning

under Microsoft Windows operating system, allows not only an accurate acquisition of

the measured profile data, but also a comparison of these data with the initial design

data introduced as IGES or DXF files. Thus there can be determined both the potential

dimensional errors and position and shape errors.

4.2 The material used and the parts realised for the experimental researches

For the experiments the author has used sheet metal blanks made of S355JR

steel according to SR EN 10025-2:2004. This is a common use construction steel with

a minimal tensile strength of 510 N/mm2, used for heavily stressed metallic

construction elements, such as masts for aerial electrical lines, runways, cranes, car

chassis, large-volume reservoirs etc. [157].

This rather cheap and unpretentious material was chosen due to the fact that the

research goal was to check the optimal configuration determined for the machine and

the control equipment and less the manner in which the machining is influenced by the

nature of the processed material. Nevertheless, as is presented in the following

subchapters, some analyses were carried out also in this regard.

38

In a first phase, of this steel type there were realised, by cutting on the oxy-gas

cutting machine OYTOME 20 HPC, from steel sheet with a thickness of 20 mm, a set of 5

square-shaped parts, with a side length of 80 mm, with a central hole of 40 mm diameter

and 5 disk-shaped parts with a diameter of 60 mm and a central hole of 8 mm diameter.

The technical limitations of the machine and of the oxygas cutting procedure,

exposed through the occurrence of relatively large dimensional errors and heat-

affected area, even after the NC system was tuned in the same manner as was later

done on the laser cutting machine, have determined however the author to give up

using this machine for more detailed experimental tests.

As a consequence, the author has decided to realise more in-depth experimental

researches on the laser cutting machine MAZAK NT-X48 Champion, described earlier,

modified by replacing the defective NC unit MAZATROL 32B with a NC unit

FAGOR 8055 taken from a NC milling machine.

The parts realised on this machine were disk-shaped, with a diameter of 60 mm

and square-shaped, with a side length of 80 mm. The disk-shaped parts had a central

hole with a diameter of 8 mm, while the square-shaped ones had a central hole of 40

mm diameter, also realised by laser cutting.

Taking into account the fact that laser cutting is suitable mostly for thinner

metal sheets, these parts were cut from 400 x 400 mm steel sheets obtained by cold

rolling, with thicknesses of 3, 6 and 8 mm respectively.

From each steel sheet type and for each part shape there were realised 18 parts

each. Thus, three different cutting regimes were used, with each one being cut 12 parts

in all (both square-shaped and disk-shaped). Three of these parts were realised under

the conditions of an untuned system, while other 9 were cut with the system optimised

through various methods, according to the results of the theoretical researches by

simulation, carried out previously and described in chapter 3 of this thesis.

Some of these parts are presented in figure 4.1.

4.3 Studies concerning the influence of the cutting regime on the

mechanical and technological characteristics

In order to better assess the effect of various cutting regimes on the material of

the parts realised within the experimental researches, these were subjected to complex

analyses, consisting in microstructural, microhardness and roughness determinations in

the area adjacent to the cut, before and after the cutting.

39

Fig. 4.1: Parts realised on the laser cutting machine MAZAK NT-X48 Champion from S.C.

COMPA S.A. Sibiu as part of the experimental researches

With regard to the roughness of the cut parts, the tests carried out within the

scope of this doctoral thesis were aimed at determining the main roughness parameters,

indicated in the standard SR ISO 4287-2001 (Geometrical specifications for products.

Surface state: Profile method - terms, definitions and surface state parameters). The

tests were done with the help of the SURTRONIC 3+ roughness tester of the company

Rank Taylor Hobson and of a specialised software for the processing of the measured

data, TALYPROF, in agreement with SR ISO 4287-2001.

Following the roughness measurements and the evaluation of results, following

conclusions could be drawn:

a) No significant differences are registered between measurements done on

various areas of a part of on different parts, cut under identical conditions. This

illustrates the constancy of reproducibility at the laser cutting of metallic materials;

b) The quality of the part's cut surface is influenced mostly by the cutting speed

and by the frequency and is insignificantly influenced by the part's thickness;

40

c) There was noticed, as expected, a worsening of the cut part's surface quality as

the cutting speed increased and the frequency decreased;

d) The studied roughness parameters Ra, Rq, vary in the same manner function of

the feed rate and frequency;

e) It can be noticed that the thickness of the cut part does not significantly

influence the quality of the processed surface.

The microhardness tests were realised on an automated microhardness tester

CV-400DAT produced by CV Instruments Europe BV and currently within the

endowment of the Centre for Studies and Researches for Metal Forming from the

"Lucian Blaga" University of Sibiu.

For the microhardness test, there was chosen one part from each set of square-

shaped parts with thicknesses of 3, 6 and 8 mm, respectively, and a test sample was

extracted by cutting with a disk milling cutter from each of these parts, with a length

of 10 mm, from the area between the outer square contour and the inner circular

contour.

The microhardness tests were of HV 0.1 type, realised with a pyramid-shaped

indenter made of diamond, with a top angle of 136°±15', driven with a load of 0.1 kgf

(0.987 N) with a penetration duration of 15 seconds.

On the polished side of each analysed test sample there were realised three

parallel rows of eight indentations each, starting from the test sample's centre towards

the outer side (the rectilinear edge), the distance between indentations being of 0.05

mm.

The increasing dimension, from the sample's outer side, which was laser cut,

towards the inner side (unaffected by the laser energy) of the indentations left by the

microhardness tester's indenter in the material, indicates that, by cutting, the part's

superficial layer was hardened, but the hardness decreased gradually with the

increasing depth. On the other hand, no significant difference between the results

achieved for different cutting regimes could be noticed.

In order to examine the specific microstructural aspects that resulted through

laser cutting, the parts were subjected to grinding and polishing, after which the

metallographic etching was done.

From the analysis of the microstructures, following conclusions could be drawn,

strengthened also by the results of the microhardness tests, carried out previously:

- the material's initial microstructure reflected on the one hand the chemical

composition, with relatively few carbon, with predominant ferritic grains (which

following the etching with nital appear in white colour) mixed with pearlitic grains

41

(which following the etching with nital appear in dark colour), and on the other hand,

through the grain arrangement on rows, it reflected the obtaining of the blank by

cold rolling;

- at the material's surface, due to the thermal effect of the laser beam, there appeared a

material layer with distinctive characteristics, of slightly lighter colour, indicating the

existence of a non-equilibrium structure, of sorbite-troostite type;

- the thickness of the heat-affected superficial layer was generally very small, of under 0.3

mm. The maximal thickness is registered for medium values of the feed rate;

- using a tuned or untuned numerical axis did not affect the material's structure in

any way, the effects of tuning being limited to the recorded dimensional and

shape errors.

4.4 Studies regarding the optimisation of processing by laser cutting on

NC machine-tools

The experimental researches described in the following were carried out on the

laser cutting machine MAZAK NT-X48 Champion, modified by adding a NC unit

FAGOR 8055 and using the set of parts described earlier. They targeted on the one

hand the determining of optimal dynamic tuning coefficients for various laser cutting

regimes and on the other hand, the validation of results obtained in this regard in the

previous stage of simulating the process with the help of the Matlab & Simulink software

package.

The cutting of 900 corners, without stopping in the corner point, represents one

of the situations in which the control system's behaviour influences significantly the

machine's contouring precision.

The experimental studies presented in this subchapter had as reference base the theoretical researches by simulation described in chapter 3.

For the beginning, the practical study targeted the influence of the feed rate on

the error in the corner point. For this, square-shaped parts, with the dimensions as

described above, were cut on the laser cutting machine MAZAK NT-X48 from S.C.

COMPA.S.A. din Sibiu, with feed rates comprised between 2000 and 500 mm/min

(0.033 - 0.008 m/s). These parts were then measured on the shape measuring machine

ZEISS CONTOURECORD 1600 D.

As an example, figure 4.2 shows the errors occurring at the cutting of a corner from steel sheets with a thickness of 3 mm with two different feed rates 2000 mm/min and 1500 mm/min, respectively.

42

Fig. 4.2: Errors measured at 90

0 corners cut from 3 mm thick steel sheet, with a cutting speed

of: a) 2000 mm/min; b) 1500 mm/min

As has been done also during the researches through simulation, the

experimental researches too have targeted the testing of various error-reduction

solutions.

Thus, in a first stage, the damping factor ξ was modified and implicitly the

amplifying factor Kpd was modified from 39.38 to 19,69. Figure 4.3 presents an

example for the cutting, with these settings, of a 90º corner from a 3 mm thick steel

sheet. It can be noticed here that the contouring errors are smaller than in the above-

presented situation.

In agreement also with the theoretical study through simulation, the experimental

researches have continued in order to identify possibilities to reduce the level of the

positioning errors even more.

Thus, a derivative component was also introduced for reducing the system's

oscillations. This component influences the contouring errors.

43

Fig. 4.3: Errors measured at 900 corners cut from 3 mm thick steel sheet, with an amplifying

factor Kpd = 19,69

Figure 4.4 presents the result of the generation of a 900 corner with different

amplifying factors on the two axes. There could be noticed a reduction of the

contouring error on the x axis.


0 corners cut from 3 mm thick steel sheet, with different

amplifying factors on the two axes

In the motion control system there was then introduced a feed-forward regulator.

As can be seen from figure 4.5, in this situation there are obtained the lowest errors,

which confirms the results of simulations carried out earlier by the thesis' author.

44


0 corners cut from 8 mm thick steel sheet, after introducing a feed-

forward controller

Similarly to the situation from the generation of the corners, there were carried out

also extensive practical studies with regard to the cutting precision, of circular profiles and

with regard to the influence of modifying some parameters and the control system's

structure on the precision.

The real contour of the obtained parts was determined with the help of the

coordinates measuring machine ZEISS PRISMO 7 S-ACC, presented earlier.

The influence factors of the dynamic behaviour of motion control systems are:

• the control parameters of the position control system on each movement axis

(represented by the amplifying factors Kpd);

• the feed rate on the axis.

Figure 4.6 presents for example the results of measuring on the coordinates

measuring machine of the circularity error for the case of generating a complete circle

with a radius of 30 mm, of S355JR steel sheets with the thickness of 6 mm, using an

amplifying factor on the position loop of Kpd = 38,38, equal for both axes, with a feed

rate v = 1500 mm/min.

45

Fig. 4.6: Errors at the generation of a circular profile for a 6mm thick steel sheet, with v=1500

mm/min

Figure 4.7 presents the case of generating in practice a full circle with

amplifying factors on the position loop Kpd = 19.69, equal on both axes, and with

cutting speeds of 1500 mm/min. There can be noticed an improvement in the

correction of circularity errors compared to the previous case.

Then, the same type of disk-shaped parts were cut, this time modifying the

amplifying factor on one axis at a time, just as it has been done also during the

simulation phase, presented in the previous chapter.

Comparing the above-presented experimental results with the ones from the

numerical simulations realised in Matlab and presented in chapter 3, it can be seen that

the experimental results confirm the previously gained results.

46

Fig. 4.7: Errors at the generation of a circular profile for a 6mm thick steel sheet, with

Kpdx=Kpdy=19,69

As a general conclusion it can be said that, while the theoretical values of feed

rates indicated by the speciality literature represent optimal values, the precision of

processed parts is significantly influenced by the parameters of the numerical axes' control

system, as shown in the previous chapters. Thus, just using an optimal feed rate, even if it

leads to an adequate roughness of the cut part, cannot compensate for the errors that occur

due to a dynamically untuned control system.

The optimal tuning of the control parameters for each machining system, based

on mathematical models and on experimental researches, influences the quality of

machined parts at least as much as a well-chosen technology.

47

Chapter 5

FINAL CONCLUSIONS AND CONSIDERATIONS. THE MAIN

CONTRIBUTIONS OF THE THESIS

5.1. Final conclusions and considerations

The theoretical researches with regard to optimising the NC cutting systems,

presented in this thesis, have targeted mainly functional aspects of these systems, but

some favourable results were obtained also through the constructive modification of

the cutting machines, with regard to using alternative components in the construction

of the motion control system of these machines.

The results of theoretical researches allow the formulation of following

conclusions regarding the realising of models used for the simulation of these systems:

The transfer function remains the most useful and most used mathematical

instrument in studying the numerical axes, a fact emphasised both by the

speciality literature and by the many software packages that allow the operation

with transfer functions;

The hybrid character of these systems, using both digital signals on the automated

position control section and analogous signals on the automated speed control

section, requires using specific mathematical methods, such as using beneath

transfer functions with complex variable “s”, specific for continuous systems also

transfer functions with the variable “z”, specific for discrete systems.

With regard to the analysed functional aspects, the study by simulation of the

rapid positioning regime allows the formulation of following conclusions:

The tuning of the position controller through the theoretical methods specific for

continuous systems does not lead to acceptable results. The system is stable, but the

dynamic behaviour is unsatisfactory;

The tuning by emulation of the numerical controller, presented in chapter 3 of the

thesis, significantly improves the dynamic behaviour of the numerical axis, but is not

sufficient. The controller's tuning through the “trial and error" method is shortened in

this way, but still cannot be avoided;

The system's dynamic behaviour significantly influences the positioning precision

on the axis.

48

The study by simulation of the contouring regime has led to following

conclusions:

When using a higher feed rate, it is necessary to modify the amplifying

coefficient of the position controller, else there appears an error with regard

to the trajectory length. Therefore, there are needed two different values of

the amplifying coefficient of the position controller, one for the positioning

regime and one for the contouring regime;

A modification of the value of the amplifying coefficient is necessary also

here, for smaller feed rates, in order to compensate with regard to the

trajectory length;

The amplifying coefficient of the position controller represents the

controllable component of the amplifying factor on the direct route. The

circular interpolation process is influenced considerably by the different

values of the amplifying factors on the axes.

When cutting arcs with angles of less than 90°, the usage of different

amplifying factors, chosen adequately, influences the contouring precision in

a positive manner, so this can be used as compensation method;

When cutting arcs with angles of more than 90°, the usage of different

amplifying factors, chosen adequately, influences the contouring precision in

a negative manner. When cutting a full circle, the resulting trajectory under

these conditions is an ellipse elongated along the direction of the axis with

the larges amplifying factor;

When processing 90° corners, without stopping in the corner point, beneath

the errors in the corner point that are due to the transitory regime behaviour,

there were noticed also errors due to the system's behaviour in stationary

regime. As compensation method the author proposed the usage of different,

adequately chosen amplifying factors on the two orthogonal axes;

As a general conclusion, both in the case of the rapid positioning regime and

in the case of the contouring regime, the system's dynamic behaviour

influences decisively its precision. The best solution for achieving both a

good positioning precision (by reducing the tracking error) and a good

contouring error (by reducing the contouring error) is the system's dynamic

compensation.

The experimental researches unfolded within the scope of the thesis had as

main goal the realising of a numerical control structure with performances specific for

a NC cutting machine. Thus, a laser cutting machine was integrated with a NC

49

equipment specific for the machining by milling, which is actually more performant

than the existing equipments for cutting machines. Thus, a constructive-functional

optimisation of the machine was realised, solving also the problems that occurred

during the integration process. The experimental validation of this solution presents

several advantages, detailed in chapter 4 of the thesis.

The results of experimental researches led to following conclusions:

The tuning of the numerical axes' controllers through analytical methods is an

extremely complex process;

The analytical expressions recommended by the speciality literature for tuning

the position controller are useful and necessary, but not sufficient. The

controller's “fine” tuning can only be done through a “trial and error” type

series of tests. This process was realised through simulation and then confirmed

experimentally, thus reducing the system's testing time and the corresponding

costs;

According to the conclusions of the simulations, the main goal of tuning the

controllers in the experimental research phase was the system's dynamic

compensation.

From the point of view of the rapid positioning regime, the experimental

researches proved following:

The conclusions of the simulations were confirmed, both from the point of view

of the quality (the simulated tuning of position controllers provided a

satisfactory dynamic behaviour of the system) and from the point of view of the

quantity (the position errors achieved experimentally were close in value to

those obtained by simulation);

The size of the minimal displacement increment influences only the values of

the position errors, these being in the order of a BLU;

Using a trapezoidal speed profile during cutting leads to the achievement of

satisfactory results.

From the point of view of the contouring regime, the experimental researches

proved following:

The experimental tuning of the position controllers confirmed the results

achieved through simulation with regard to the best method for tuning the

controller;

The errors of closing the circular contours were relatively close to the ones

determined through simulation, namely in the order of one minimal

displacement increment.

50

5.2 Contributions of the thesis

Through the researches carried out and presented concisely in this thesis ((some

of the more important aspects being presented in more detail through the publication

of scientific papers), a series of original contributions were brought to the tackled

domain. In brief, the main contributions are as follows:

With regard to theoretical researches:

- Elaboration, based on selective bibliographical researches, of an analysis on the

state of the art and the perspectives of the precision of NC machine-tools and the

identification of numerical axes as main element in this regard;

- Elaboration of a mathematical model, with elements of originality, for

numerical axes using as execution element the DC servomotor, taking into

account the influence of performance-limiting factors;

- Introducing in the models' structure of the influence of numerical, computer-aided

control (discrete transfer functions, sampling periods, numerical controllers);

- Elaboration of mathematical models for the generation of reference parameters, both

in rapid positioning regime and in linear and circular interpolation regime;

With regard to experimental researches:

- Tuning of the controllers of the studied numerical axes, both based on the results of

simulations realised on theoretical models and based on “trial and error” procedures;

- Theoretical and experimental research and validation of solutions for the

improvement of the dynamic behaviour and kinematic precision;

5.3 Directions for continuing the researches

As has been shown also in the previous chapters, the results achieved within the

present thesis have a high potential of being applied in the industry. This fact allows

also for extension possibilities of the theoretical and experimental researches.

Thus, among the possible directions for continuing the researches, there can be mentioned:

- realising mechanical and control structures that allow the generation of spatial

trajectories (spatial contouring - simultaneous control of 3 or more axes);

- implementing interpolation algorithms with superior performances (spline-type

interpolation algorithms);

- replacing the control algorithms used so far (PID) with adaptive control

algorithms, fuzzy and neuro-fuzzy control algorithms, introducing control

algorithms based on neuronal networks etc.

51

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