MECHATRONIC ANALYSIS OF MACHINE TOOLS

MECHATRONIC ANALYSIS OF MACHINE TOOLS

Ramon Maj, Giacomo Bianchi Institute of Industrial Technologies and Automation

National Research Council Viale Lombardia 20/A 20131 Milano, Italia

[email protected] ; [email protected]

Abstract: High speed machine tools show close interaction between dynamic behavior of the mechanical structure, drives and numerical control. In order support designers of high performance machines, new analysis tools for an integrated holistic mechatronic optimization have been developed in the European project MECOMAT. Fully parameterized SAMCEF models of the mechanical structure are first optimized using BOSS quattro V4, drivers built in SIMULINK are linked with SAMCEF Mecano and tuned. Subsequently different analyses have been performed to evaluate the performances of the complete mechatronic system.

INTRODUCTION

Today market is asking for fast and accurate machine tools, to reduce machining time and assure the required precision, constructed with stiff but light mechanical structures, and drives with wide bandwidth. Traditional machine tool design is performed by separated and sequential optimization of the mechanical structure and the control system. Today this approach becomes less and less effective because rigid transmissions and light structures are often used, requiring to consider also the distributed compliance of the structure while evaluating the controlled system performance. Therefore it’s essential to study the behavior of the structure with the control system active, using an integrated mechatronic approach that considers the dynamic coupling between them [1]. Many European (and Italian in particular) machine tool manufacturers offer a high number of machine variants, each one customized for the specific needs of a reduced number of customers: each product version is usually obtained starting from existing solutions and modifying some key geometrical dimension or structure size. In this scenario, the whole product range can be usually categorized in a small number of basic “families”, based on the same “machine concept” (i.e. typical applications, axis disposition, transmission typology, etc.) and, sometime, even different families share some common components, following a modular approach to minimize production costs and maximize product quality. When a new machine has to be developed the “product concept” has first to be defined, requiring the evaluation of several design variants in a short time: a very fast assessment is usually possible only if a specific design experience on similar machine is available, otherwise the time required to estimate the performance of the mechanical structure is usually too long. The MECOMAT project, “MECHATRONIC COMPILER FOR MACHINE TOOL DESIGN” [2], was finalized to develop tools and methodologies to

9th SAMTECH Users Conference 2005 1/16

support machine design, starting from the conceptual phase, by mechatronic synthesis, analysis and optimisation. The present paper presents the modeling approach developed by ITIA-CNR in the mentioned project, applied to a three axis machining center, proposed as a demonstrator by the COMAU company.

1 ANALYSIS SUPPORT FOR THE “PRELIMINARY DESIGN” PHASE

After the best machine architecture has been selected during the conceptual design phase, in the next step, often called “preliminary design”, the machine is defined more in detail, identifying optimal values for a large set of continuous and discrete parameters, like geometrical dimensions, structure thickness, number of ribs, etc. (see Figure 1). Many constraints and goals, of different nature, have to be taken into account, like stiffness and weight specification, manufacturability, cost, etc. It is important to support this phase with analyses performed by different CAE tools, for example to evaluate the behavior of the mechanical structure. In the industrial practice the first structural models often represent the structure as a set of rigid bodies connected by guideways modeled as lumped compliances. This approach is useful for a first comparison between alternative layouts, but, especially for high speed machines, the approximation done is often too large, because the goal of inertia minimization produce light structures with non negligible distributed compliance.

.

Figure 1 A typical machine tool structure, with stiffening ribs

The development of Finite Element models (“FEM”) of the structure is today quite time consuming and, therefore, is relegated to later development phases, when a very limited number of design variants is under evaluation. FEMs are typically used to evaluate purely mechanical aspects, like the static stiffness at the tool or the first natural frequency with locked motors: an indirect estimation of performances like machining capability and motion accuracy is possible only thanks to the large designer experience,


built on similar machines evaluated in the past. This approach is usually very effective but becomes quite risky when especially innovative design solutions, like machine tools adopting linear electrical motors and/or a parallel kinematics concepts: the relationship between purely mechanical analyses and machine performance is less known. Nevertheless it has to be noted that even revolutionary machines can be evaluated by an expert end-user, performing several standard or custom tests, from tracking accuracy along specific trajectories, to part machining and subsequent measurement. Starting from this consideration, the “Virtual prototyping” approach has been developed [1]: the basic idea is to build numerical models that permit to simulate some of the most significant tests performed in practice. To reach this goal is necessary to build a model able to execute a part program and containing a sufficiently accurate description of all phenomena influencing the quality measure related to the selected tests. Following mentioned considerations, in the framework of the MECOMAT project a specific modeling approach has been proposed for the “preliminary design” phase, with the following goals:

1. quick, even if approximate, structural analysis of the machine;

2. evaluation of a machine family, defined by a parametric model, enabling sensitivity analysis and optimization, to highlight the most critical regions in the structure, that will require an accurate stiffness and/or mass optimization in the following activity (“detailed design”)

3. preliminary evaluation of the machine in operating conditions by the development of a “virtual prototype”, joining the structural model to a representation of the control system;

The proposed methodology doesn’t pretend to generate “automatically” a detailed design of the mechanical structure: the designer knowledge remains essential, also to correctly evaluate numerous constraints like maintainability, ergonomics, production cost, legislation compliance, etc. The goal here is permit a fast and more accurate analysis of several machine variants. To evaluate machine accuracy during finishing operations with high acceleration (e.g. a milling machine finishing the sculptured surface of a mould), it is useful to execute mechatronic simulations of motion with no cutting force (avoiding to incur in the complexity of cutting process modeling) as done in industry, using external instruments to measure the tool position (starting from the circular trajectories specified by the ISO 230-4 standard [3]) or performing very light machining operations along specific trajectories and reconstructing the tool position from the workpiece surface profile. In order to simulate machine motion it is necessary to describe in the model, besides the mechanical structure, also components like motors, sensors, drives and control system. This kind of models can be very complex, because it is necessary to correctly reproduce the characteristics of all components that can limit machine performances. For an effective industrial application of the mechatronic modeling it is therefore required to have at disposal an appropriate library of component models. This library can be built by research centers and large machine tool companies (as it is sometime the case today)


or supplied by the corresponding industrial suppliers of the “physical component” (a very rare case until now). A component for which the collaboration with the corresponding supplier is fundamental is the Numerical Control (“NC”), because it would be extremely complex and unreliable to build a self-made model reproducing the trajectory generation strategy adopted by each specific NC, starting from the part program in ISO format. At the moment some producers of NC are able to supply SWs that reproduce on a standard PC, perhaps with some limitations, the reference position signals calculation (NUM [4], [5] and FIDIA [6]).

2 MACHINE MODELING FOR PRELIMINARY DESIGN

To achieve the previous objectives, models must represent correctly the behavior of the key machine components. Models have been built exploiting the SAMCEF environment [7] but also Matlab-Simulink1 [8] (and “Real Time Workshop” to generate the corresponding C code) and external code, for specific analysis and components.

2.1 Structural modules

Different approaches were considered, with different accuracy and technical difficulty. In this paper we describe an approach based on SAMCEF fully parametric structural FE models, with shell mesh. A detailed model has to be defined for each structural module considered. Internal ribs distribution and Finite Element automatic meshing are controlled by specific parametric formula.

2.2 Guideways

The model is built with SAMCEF elements and can be used for linear recirculating guideways. The model is able to represent kinematics, compliance and friction of the guideway.

2.3 Kinematics chain

The proposed model built is built with SAMCEF elements and describes an axe transmission based on a recirculating ball screw, with the corresponding bearings and coupling connecting the electrical motor. A specific MECANO element has been developed during the project to represent a screw joint acting along a flexible beam representing the screw.

2.4 Sensors

The control loop is usually based on the actual position measured by a linear optical encoder. The model is built for the mechanical part with SAMCEF elements (and it is assembled, together with the kinematic chain, in the correct location on the machine), representing local resonances and play, if desired. A Simulink model can be added, to reproduce the measuring error and bandwidth limitation.

1 Matlab Simulink and Real Time Workshop are trademarks of The MathWorks, Inc.


2.5 Drives and velocity regulators

The drive model is built in Matlab-Simulink. Different models have been developed taking as reference industrial products [10]. In this paper we present a simplified model reproducing the classical structure with concatenated velocity and position loops.

2.6 Numerical Control

The Numerical Control model must be able, as done in reality, to interpret an ISO part program and generate the position references for each machine axis. This operation is differently done by each NC manufacturer and constitutes a strategic industrial know-how. To correctly reproduce the behavior of a specific NC, without diffusing the proprietary algorithm, the NC producer is invited to provide a “black-box” routines (possibly simply extracted from the real NC code in case of PC-based NC), that performs the calculation without revealing confidential information. Also NC installed on real machine tool could provide the same information (saving to a file the requested signals), but this solution would slow down considerably analysis and optimization of several machine variants.

2.7 Structural Model of the machine

Using a SAMCEF fully parameterized FE model for structures, kinematic chains and guideways, we will be able to represent all the structural parameters taken into account during the preliminary design process. The machine used as test case is a three axis milling machine built with box-in-box architecture and a gantry X axis.

TOOL TIP

RAM

SLIDE Y

GUIDEWAY Z

TOOL TIP

RAM

SLIDE Y

GUIDEWAY Z

Figure 2 –Model of Y and Z machine axis with component nomenclature.

The parameterized model permits to perform structural optimization, imposing objectives and bounds and varying geometrical dimension or other structural parameters like guideway stiffness, ball screw diameter etc. Each component model is described by a large number of parameters, permitting to explore very different design solutions for the machine, belonging to the selected “Machine Family”. The designer can specify the working volume (or the corresponding axis strokes) and performances requirements (as static stiffness).


Figure 3 –Model of the complete machine with component nomenclature.

FIX PART

SLIDE Y

MOTOR X1

GUIDE-WAY X/Y

MOTOR Z

MOTOR YDRIVE Y

DRIVE Z

DRIVE X1

RAM

MOTOR X2

DRIVE X2

SLIDE X

FIX PART

SLIDE Y

MOTOR X1

GUIDE-WAY X/Y

MOTOR Z

MOTOR YDRIVE Y

MOTOR YDRIVE Y

DRIVE Z

DRIVE X1

RAM

MOTOR X2

DRIVE X2

SLIDE X

Figure 4 –SAMCEF mechanical model of the machine with different parameters


2.7.1 Guideway

The proposed model describes linear recirculating guideways, reproducing the kinematics, compliance and friction [11]. The model includes also some structural parts, used to stiffen the machine structure where guideways are located to avoid excessive local deformations. In order to be able to automatically assemble the whole machine model while the guideway location is varied, changing the corresponding parameters, it has been decided to consider the stiffening elements as part of the guideway model: during the assembly process they will be “glued” to the corresponding structural elements (by the .STICK command).

Ribs: shell 4 nodes

MEANelement

Rail: Beam element

Flexible slider

Carriage: spring between 2 coincident

nodes

stiffening plates and carriage

Ribs: shell 4 nodes

MEANelementMEAN

element

Rail: Beam element

Flexible slider

Flexible slider


nodes


nodes

stiffening plates and carriage

FIX part

SLIDE X GUIDEWAY X

Figure 5 SAMCEF model of guideway

2.7.2 Ball screw kinematic chain

To build a typical machine tool kinematic chain based on a ball-screw, the following model, based on new SAMCEF element Flexible screw, has been proposed:

Third nodeto drive

SCRF

stiffeningelements

MEAN

stiffening

MEANnut

node

COUPLINGJOINT

ROTOR

STATOR

BEAM

STICK

BEARING

elements

stiffeningelements

MEAN

STICK

STICK

Third nodeto drive

SCRF

stiffeningelements

MEAN

stiffening

MEANnut

node

COUPLINGJOINT

ROTOR

STATOR

BEAM

STICK

BEARING

elements

stiffeningelements

MEAN

STICK

STICK

Third nodeto drive

SCRF

stiffeningelements

MEAN

stiffening

MEANnut

node

COUPLINGJOINT

ROTOR

STATOR

BEAM

STICK

BEARING

elements

stiffeningelements

MEAN

STICK

STICK

Figure 6 Scheme of ball screw kinematic chain model


The model is able to represent - Ball screw kinematics, with nut compliance and friction - Support bearing local compliance - Motor mass, inertia and screw coupling compliance - The kinematic chain model includes also the position sensor

2.7.3 Drive

1Out1

s

den(s)

Vel by approxderivation1

s

den(s)

Vel by approxderivation

Kp_VKp_P

Ki_V

1/s

Integrator

Feed_Forward

Acc_Forward

8Act_VEL

7Act_POS

6POS_REF

5AXIS_Inetria

4FF_Gain

3VEL_P_Gain

2VEL_INTR_Gain

1POS_P_Gain

Figure 7 Simulink model for axis drive

A simplified version of an industrial Drive is proposed, with an inner PI (Proportional and Integral) velocity loop and an outer P position loop. Velocity and acceleration Feed Forward with limited bandwidth derivation are present. This model has been compiled with Real Time Workshop and linked with MECANO and it can be used for non linear time simulations.

2.8 PRELIMINARY DESIGN OPTIMIZATION

With the machine parameterized model it is possible to perform different kinds of analysis. First linear structural analyses were performed during an optimize process driven by BOSS quattro V4, mimicking the process of the mechanical designer to optimize the structural parameters. The optimal solution was then inserted in the complete mechatronic model of the virtual machine.

2.8.1 Structural preliminary design optimization

Structural preliminary design is here optimized evaluating the static stiffness at the tool and normal modes with locked motors. The imposed bounds are minimum value for static stiffness and maximum value for the moving masses. The objective is to maximize the first resonant frequency (locked axes). Figure 8 describes the optimization study definition in BOSS quattro V4.



1 2

K_x > 35N/µm

K_y > 35N/µm

K_z > 100N/µm

Maximize fir st natura l frequency

Axes MASS bounded

Figure 8 – BOSS quattro V4 scheme for structural optimization.

Bounds Static stiffness in X Kx > 35N/µm Static stiffness in Y Ky > 35N/µm Static stiffness in Z Ky > 100N/µm Axe X Mass Mx <1500kg Axe Y Mass My <660kg Axe Y Mass Mx <380kg

Figure 9. In Figure 10 represents objectives and bounds variations during iterations in an optimization of some geometrical parameters in SLIDE_X and RAM. This is a simple but significant example of what could happen in an engineering department during the design process of a new machine tool.


Figure 9 – Design Parameters variation during an optimization.

Figure 10 – Stiffness bound and frequency objective variation during an optimization.

RAM External thickness

RAM Internal ribs thickness

EXT TICK

INT TICK

K_y >35N/µm

Maximize First natural frequency

K x >35N/µm

2.8.2 Super Element Reduction and Drives Tuning

After optimizing the structures, a reduced order linearized model has been generated, adopting the component-mode synthesis approach [9]. The non linear MECANO model is linearized and passed to DYNAM that performs the Super-Element (“SE”) reduction. The following “boundary” or “connection” dofs where selected: motors third nodes and tool translations. The linearized model is imported in Matlab-Simulink environment to perform Drives Tuning, using frequency domain analysis.

Figure 11 – Simulink model of mechatronic system.

Tool_X_Vel

Sine Wave2

Sine Wave1

Sine Wave

-K-

Drive_Z_Vel

-K-

Drive_Z_Dep

POS_REFAct_POSAct_VEL

Force

Drive_Z

-K-

Drive_Y_Vel

-K-

Drive_Y_Dep


Force

Drive_Y

-K-

Drive_X2_Vel

-K-

Drive_X2_Dep


Force

Drive_X2

-K-

Drive_X1_Vel

-K-

Drive_X1_DepPOS_REFAct_POSAct_VEL

Force

Drive_X1

iDrive_X1

iDrive_X2

iDrive_Y

iDrive_Z

iTool_X

iTool_Y

iTool_Z

oDrive_X1_DEP

oDrive_X2_DEP

oDrive_Y_DEP

oDrive_Z_DEP

oTool_X_DEP

oTool_Y_DEP

oTool_Z_DEP

oDrive_X1_Vel

oDrive_X2_Vel

oDrive_Y_Vel

oDrive_Z_Vel

oTool_X_Vel

oTool_Y_Vel

oTool_Z_Vel

COMAU_MF1Mod

Tool_X_DEP

Driv e_X1_De

Driv e_X2_De

Driv e_Y_De

Driv eZ_Dep

Tool_Y_DEP

Tool_Z_DEP

Driv e_X1_V

Driv eX2_Ve

Driv e_Y_V

Driv e_Z_Ve

Tool_Y_Vel

Tool_Z_Vel

Frequency (Hz)

0

0

0

0

0

0

0

0

0

100

101

102

0

5

0

0

0

Velocity X/Ref XFREQUENCY domain analysis

(Hz)Frequency

0

0

0

0

0

0

0

0

0

100

101

102

0

5

0

0

0

Velocity X/Ref XFREQUENCY domain analysis

TUNING PARAMETERS

Matlab-Simulink is a powerful environment to perform drives tuning. Frequency or time domain analysis are quickly performed using linearized models of the structure, permitting to reproduce some tuning procedures used on real machines.

2.9 COMPLETE MECHATRONIC MODEL OF THE MACHINE

The mechatronic model of the machine is build in the SAMCEF environment. The drives are linked with MECANO and the axis references for time simulations are read from data files generated by a NC emulator provided by NUM. Once tuned the drives, the model can be used as a Virtual Machine Tool (“VMT”), to perform time simulations, like on real machine tests. We can evaluate the effect of different parameters, relating to sensors, drives, tuning parameters and NC. The VMT on one side is an approximate representation of a real machine, but, on the other side, permits a very deep study of its behavior, that would be possible on a physical prototype only adopting a very complex and costly instrumentation system, able to measure machine motion and deformation during the experiments.


Samcef-Simulink Drives Models

MOTOR X1

MOTOR Z

MOTOR X2

DRIVE X2

DRIVE Z

DRIVE X1

Motor torque

ISO PART PROGRAM N100 G1 X0 Y100 Z0 F120000 N110 G17 G2 X50 Y50 I0 J50 N120 G17 G3 X90 Y10 I90 J50 N130 G17 G2 X90 Y0 I90 J5 N140 G1 X0 Y0 M2

AxisReference

s

NUM CN

EMULATOR

SAMCEF Mechanical Model

MOTOR Y

DRIVE Y

DRIVES TUNING

PARAMETERS

Axis Sensors

Figure 12 – SAMCEF mechatronic model of the machine .

2.9.1 Static Stiffness evaluation with active control

It’s possible to compute deformations due to static forces applied on tool, uniform axis accelerations etc . The mechatronic model permits to reproduce a typical Experimental Static Stiffness Test, evaluating also the effects of linear sensor location and drives tuning on static stiffness at the tool and disturbance rejection.

0 0.5 1 1.5 2 -20

0

20

40

Time (s)

Static Response in X direction

Tool Force/20 (N)Sensor Deplacement (micron)Tool Tip Deplacement (micron)

Figure 13 – Time simulation of Static Stiffness test.

2.9.2 Tool Tip Impact Test

The dynamic compliance at the tool can be evaluated computing the corresponding Frequency Response Function (“FRF”) with active drives. Like in experimental test,


time simulation are performed imposing a stand-still command to the axis and applying an impact force on tool (see the following figure).

-0.05

0

0.05

0.1

0.15 Tool tip time response in X direction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1

0.15

Tool

dis

plac

emen

t (m

m)

Time (s)

Locked Motors

Driven Axis

0 0.0025 0.005 0.0075 0.01

5000 Tool TipImpact Force

Time (s)Forc

e (N

)

Figure 14 – Time simulation of impact test.

Time simulation results are then elaborated to in Matlab to obtain the corresponding information in the frequency domain. It is therefore possible to investigate the effect of tuning parameters on the dynamic compliance tool FRF.

10 100 400-190

-180

-170

-160

-150

-140

-130

-120

Frequency (Hz)

Amplitude (dB)

FRF at Tool tip in X direction

Locked motorsDriven Axis

Figure 15 – Comparison between dynamic compliance at the tool with locked motors

and active drives.


10 100 400-190

-180

-170

-160

-150

-140

-130

-120

Frequency (Hz)

Am

plitu

de (d

B)

FRF at Tool tip Driven Axis

X DirectionY Direction

Figure 16 – Dynamic compliance at the tool in X and Y direction with Active Drives.

2.9.3 Trajectory Execution (without cutting forces)

After having tuned the axis, time simulation can be performed imposing as a reference position on the axis the required trajectory, generated by the NC Emulator. As example the execution of a square angle with 20mm side is shown, using different Acceleration Time values (Tacc: it is time that the NC uses to reach the maximum acceleration). The ISO instruction are G1 X20 Y0 Z0; G1 X20 Y20.

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

Velo

city

[m/s

]

Axis Velocity along Trajectory Tacc=5ms

0 0.05 0.1 0.15 0.2 0.25-10

-5

0

5

10

Time [s]

Acc

eler

atio

n [m

/s2]

Axis Acceleration along Trajectory Tacc=5ms

Acc Axe XAcc Axe Y

Vel Axe XVel Axe Y

CicleTime=0.25s

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05

0.1

0.15

0.2

0.25

Velo

city

[m/s

]

Axis Velocity along Trajectory Tacc=50ms

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-4

-2

0

2

4

Time [s]

Acc

eler

atio

n [m

/s2]

Axis Acceleration along Trajectory Tacc=50ms

Acc Axe XAcc Axe Y

Vel Axe XVel Axe Y

CicleTime=0.35s

Figure 17 –Velocity and acceleration profiles from the NC emulator.

Adopting Tacc = 5ms and Tacc = 50ms, different velocity and acceleration profiles are obtained, as shown in Figure 17. This produces different performance on trajectory execution. The trajectory with Tacc =5ms is faster but less accuarte than the trajectory with Tacc =50ms as shown in the following figure.


18 18.5 19 19.5 20 20.5 21 21.5 22

0

2

4

6

8

10

X Displacement

Y D

ispl

acem

ent

Trajectory references, sensors and tool

Reference T acc=5msSensor T acc=5msTool T acc=5msReference T acc=50msSensor T acc=50msTool T acc=50ms

Figure 18 Trajectory execution. Reference position, compared to the position measured

by the linear sensors and to the actual tool position.

3 Conclusions

The paper presented a modeling and analysis methodology developed during the MECOMAT project. The main goals were:

1) Fast model development and analysis, compatible with the short timing of customized product development in the machine tool sector

2) Mechatronic modeling, in a “Virtual Prototyping” prospective, to evaluate innovative design solution for high speed machines, where structure-control interaction is significant

The proposed approach is based on the development of parametric models, describing design solutions belonging to a set of “machine families”. While the development of mechatronic models is today basically possible with the most advanced CAE packages, an efficient industrial application in sectors where, as for machine tools, the analyzed product contains a lot of sophisticated commercial components, will be possible only when an adequate library of numerical models will be commercially available.


4 Acknowledgements

The authors would like to acknowledge the European Commission for sponsoring the MECOMAT project in the ‘Competitive and Sustainable Growth’ Programme (N°: GRD1-2000-25270, 2001-2004) and the valuable work of all project partners (CE.S.I. (I), K.U. LEUVEN R&D (B), LANCASTER UNIVERSITY (UK), BUDAPEST UNIVERSITY (HU), ITIA-CNR (I), CETIM (F), COMAU (I), HOLROYD (UK)).

5 References

[1] G. Bianchi, F.Paolucci, P. Van den Braembussche, H. Van Brussel, “Towards Virtual Engineeloop in Machine Tool Design”, Annals of CIRP, Vol. 45/1/1996

[2] MECOMAT project “MECHATRONIC COMPILER FOR MACHINE TOOL DESIGN” (GROWTH N°: GRD1-2000-25270); Final Publishable Report

[3] “Acceptance code for machine tools, Part 4: Circular tests for numerically controlled machine tools”; ISO/DIS 230-4; 1994

[4] NUM: Schneider Electric –NUM P.O. Box 68 – 95101 Argenteuil, Francia, CNC production http://www.schneider-num.com

[5] R. Bearee, P.J. Barre, S. Bloch ; "Influence of High-Speed Machine Tool Control Parameters on the Contouring Accuracy. Application to Linear and Circular Interpolation”, Journal of Intelligent and Robotic Systems; 40 (3): 321-342, July 2004; Kluwer AP

[6] FIDIA: FIDIA S.p.A. -Corso Lombardia, 11; 10099 San Mauro Torinese (TO) – ITALIA http://www.fidia.com

[7] SAMTECH Rue des Chasseurs-Ardennais, 8v B-4031 Liège (Angleur), BELGIUM. http://www.samtech.fr

[8] Matlab, Simulink: “The language of Technical Computing”; The MathWorks, Inc. http://www.mathworks.com

[9] Craig R.R., Bampton M.C.C, Coupling of Substructures for DYNAMIC Analysis, AIAA Journal, Vol 6, No 7, July 1968, pp. 1313-1319

[10] Siemens: SIMODRIVE 611 digital SINUMERIK 840D/810D: Description of Functions. Drive Functions. Codice 6SN1 197–0AA80–0BP7

[11] W. Symens, F. Al-Bender, J. Swevers, H. Van Brussel; “Dynamic Characterization of Hysteresis Elements in Mechanical Systems”, American Control Conference 2002; IEEE1192


http://www.schneider-num.com/

http://www.fidia.com/

http://www.mathworks.com/

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MECHATRONIC ANALYSIS OF MACHINE TOOLS