Optimal control laws for chatter suppression using inertial actuator in milling processes

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Optimal control laws for chatter suppression using inertial actuator in milling processes

Iker Mancisidor, Jokin Munoa, Rafael Barcena

To cite this version:

Iker Mancisidor, Jokin Munoa, Rafael Barcena. Optimal control laws for chatter suppression us-

ing inertial actuator in milling processes. 11th International Conference on High Speed Machining

(HSM2014), Sep 2014, Prague, Czech Republic. �hal-01134192v3�

OPTIMAL CONTROL LAWS FOR CHATTER SUPPRESSION USING INERTIAL ACTUATOR IN MILLING PROCESSES

I. Mancisidor¹*, J. Munoa¹, R. Barcena²

1Dynamics and Control, IK4-Ideko, Elgoibar, Basque Country, Spain

2Department of Electronic Technology, University of the Basque Country (UPV/EHU), Bilbao, Basque Country, Spain

*Corresponding author; e-mail: [email protected]

Abstract

Actually, active control devices are considered as one of the most suitable chatter suppression methods for heavy duty machining operations. Generally, they are based on the generation of a reaction force over the main structure, controlled by a feedback control law, which depends on the real time vibration measurement. Several active control strategies have been proposed in the literature and due to the limited space offered by the machines for these actuators, the active control system has to be optimized, including the control law. In this work, the most employed strategies will be compared by means of a verified mechatronic model and the optimal law will be sought for each zone of stability lobes. Finally, experimental tests will be performed for verifying the results obtained by the mechatronic model simulations.

Keywords:

Chatter; Active Control; Inertial Actuator

1 INTRODUCTION

The presence of self-excited vibrations, known as chatter, in machining processes is a classic problem that limits the productivity. These vibrations, caused principally by the regenerative effect [Tlusty 1957; Tobias 1958], prevent obtaining required surface finishes and decrease the life of tools and mechanical components of the machine.

The location of a tuned vibration absorber in the structure is a widely studied solution for increasing the stability margin [Den Hartog 1956; Sims 2007]. However, the employment of a passive absorber is not feasible in processes where the dynamics of the machine change according to the working position during machining.

Therefore, the integration of active dampers has been widely analyzed for many years, as one of the most effective chatter suppression methods, due to their adaptability to variable conditions. In 1970, Cowley and Boyle proposed to use an electromagnetic inertial actuator with an accelerometer in order to introduce active damping into the structure of a machine tool [Cowley 1970]. Some years later, Ehmann and Nordmann [Ehmann 2002]

recommended introducing inertial actuator in large machine tools with vibration problems.

Among inertial actuators, electromagnetic devices are usually employed [Loix 2004; Sabalza 2004; Baur 2012], although some hydraulic actuators have been proposed in the literature [Brecher 2004; Brecher 2012]. However, the hydraulic actuators present several disadvantages, such as time delays to reach the required pressure, leak oil,

hysteresis effects, maintenance problems or other nonlinearities. These issues make the electromagnetic to be the most employed technology to develop inertial actuators.

When these active control systems are designed, one of the most important parameters is the ratio between the force capability and the occupied volume. The reason is that generally machine tools offer reduced spaces for the actuator location close to the tool and, hence, their volume has to be reduced. Therefore, the performance of the actuator has to be exploited to the fullest, which depends partially on the active control strategy employed for calculating the force. Therefore it is important the optimization of the performance of the control law.

Several active control laws have been proposed in the literature, which can be classified under two general categories [Preumont 2002]: feedback and feedforward.

Generally, feedback active controls are applied in machine tools, since feedforward controllers require a reference disturbance signal, which is very complicated to achieve when it is referred to chatter vibrations. In a typical feedback control loop, a sensor will measure the total response of the mechanical structure, which is then fed via

Fig. 1: Chatter vibration scheme with feedback control.

the controller to the secondary actuator, creating a mechatronic system (see Fig. 1).

Among feedback control laws, some authors have proposed algorithms which require the knowledge of the system behavior based on a mathematical model. Some of them, such as Virtual Passive Absorber (VPA) and Virtual Passive/Active Absorber (VPAA) have been designed for chatter suppression [Huyanan 2007; Bilbao- Guillerna 2010]. These strategies can benefit the control since efforts of actuators can be focused on the modes or frequencies of interest, not wasting over all vibrations.

However, in some cases, the elaboration of a mathematical model can be a complicated task and, at best, can only be a low-dimensional approach of the actual system. Moreover, for certain machine tools, the model should be modified for each position of the tool over the workspace.

Model free algorithms have the advantage that the reduction of vibrations can be achieved without a model of the structure and, hence, they can adapt their behavior in cases where the dynamic properties are changing inside the workspace. Therefore, they are prepared to control a wide range of frequency, which is usually limited by a high-pass and a low-pass filter introduced by the user.

The objective of typical feedback control laws found in the literature is to reduce the vibration of a mechanical system by automatic modification of the system’s structural response [Fuller 1996]. In this way, the modification of the main dynamic parameters has been sought using position, velocity and acceleration feedback [Brecher 2004].

Recently, a new model free control strategy, called Delayed Position Feedback, has been proposed [Munoa 2013]. Such control law is based on the reduction of the regenerative effect and theoretically it can improve the stability.

One of the most important effects which limits the performance of feedback controller in mechanical systems is the unmodelled system delay. Such phase shift may arise because of the dynamic response of the sensors and actuators being used, or may be due to time delays in the controller. If either position or acceleration feedback are implemented in a control loop with a system delay, the effect of the delay changes considerably the damping of the system [Fuller 1996]. This fact modifies significantly the behavior of the control and, in some cases, the system can become unstable. In contrast, when velocity feedback is employed, the delay has only a small effect on the effective mass and stiffness and, hence, it is thus seen to be a more robust control strategy. Meanwhile, when the novel strategy (Delayed Position Feedback) is employed, the force is introduced with the tooth passing period delay, which is usually higher than the existing delays due to the actuator and controller. Therefore, these undesired delays can be compensated, reducing the tooth passing period delay of the feedback control.

The present work performs the comparison between these four model-free feedback control strategies, by means of the milling mechatronic model [Munoa 2013]. In the first part, their mathematical explanation is provided briefly and

Fig. 2: Points taken into account in the mechatronic model.

then, they are compared theoretically. Finally, they are implemented in a real cutting process, showing which control strategy is the optimum for each zone on a stability lobe.

2 MATHEMATICAL FORMULATION

This section presents the mathematical development of the analyzed control laws. Both the frequency domain and time domain formulations can be employed in order to study the stability of a cutting process [Munoa 2009]. The regenerative effect is clearly explained by frequency formulations, although non-linearities cannot be considered [Zatarain 2004; Stepan 2011].

Therefore, the present work has considered the formulation of the mechatronic model [Munoa 2013]. Such mechatronic model simulates the cutting process, where the influence of inertial actuator control loop is integrated into the regenerative effect. It is based on time domain simulations, which permit the inclusion of non-linearities in its formulation. In this way, such model can be employed to simulate easily the possible limitations and saturations of the inertial actuators, such as force or stroke saturations.

2.1 Time domain formulation

The present subsection summarizes the formulation employed by the mechatronic model in order to simulate the milling cutting process with active inertial actuator effect.

The general time domain equation is considered,

  t C r   t K r   t F

_c,p

  t F

_act,q

  t r

M          

(1) where r={rp, rq, rm}^T. The subscripts mean the location of each parameter, where p is the cutting point, q is the actuator location and m is the measurement point, for cases where the actuator and sensor are not collocated (see Fig. 2).

Following the linear formulation [Altintas 2012], the cutting forces can be defined as,

 

^































 













) ( sin

) sin(

1

T t

f K K b K F

F F

p j

z ac rc tc

a,j p r,j t,j

r u Δ





 

j ae

re te

g S K K K



























 ⁽²⁾

If the tangential, radial and axial forces (Ft, Fr, Fa) of the inserts are projected onto the xyz Cartesian axes:

  _{ }

^

 











 



 

 





 



 



 

 





¹

0 1

0 , , , ,

c

,

Z

j

a,j p r,j t,j j

Z

j j p z

j y

j x p

F F F

F F F

t β  

F

⁽³⁾

where

 























































sin cos

0

) cos(

cos ) cos(

sin )

sin(

) sin(

cos ) sin(

sin ) cos(

, _{j j j}

j j

j

β j

The actuator force Fact performed in point q, depends on the selected control law and is analyzed in the next subsection.

Assuming proportional damping, if the modal transformation is performed, the equation (1) is as follows:

 

t C'η

 

t K'η

 

t Q_p^TF_c,p

 

t Q_q^TF_act,q

 

t η

M'       ⁽⁴⁾

being M’=Q^TMQ, C’=Q^TCQ and K’=Q^TKQ, the normalized and diagonalised modal mass, damping and stiffness, respectively and Q={Qp, Qq, Qm}.

Then, it can be simplified for each mode,

       

i i i i i i i

i m

t t P η η t

η t 2

 

_n,  



_n,²  ⁽⁵⁾ where mi, ωn,i and ξi are the modal mass, the natural frequency and the relative damping ratio of ith mode of structure, while Pi is the modal force.

Modal displacements are calculated numerically based on state space formulation [Bediaga 2009]:

     

   

i i i

i i i i i

i

m t P η t

η t η t

η t



 















 







 









1 0 2

1 0

, n 2

,

n 













⁽⁶⁾

     

   

i k i i i k

i k i i i

i i k

i k i

m t P t t

t η t

η t A A

A A η t

η t































 

















2 , n 2

, 22 , 21

, 12 , 11 1

1 2



 





 ₍₇₎

Then, the Cartesian displacements are calculated by the modal vector matrix and they will be used in the next step.

 

t_k1 Q^Tη

 

t_k1

r ⁽⁸⁾

In this way, the results can be showed in graphs. In particular, Fig. 3 shows the results for a chatter-free and for a chatter case.

Fig. 3: Results obtained by the time domain model; a) Time domain signal of a chatter-free case; b) Spectrum of the speed of a chatter-free case; c) Time domain signal of a chatter case; d) Spectrum of the speed of a chatter case.

2.2 Model-free control laws

This subsection analyses the most employed control laws when inertial actuators are used for chatter suppression.

Four control strategies have been selected considering different terms of the general regenerative dynamic equation (4).

In order to show clearly the effect obtained by each control algorithm, the system is analyzed in frequency domain. In such formulation, the zero order approximation has been considered.

When the stability of the linear system is analyzed in frequency domain, only the dynamic part has influence, so equation (2) can be simplified as

 

¹

 

^{, 1 () ( )}

0

T

c,



^

 



































^Z

j

j p

ac rc tc

p j t g

K K b K

t β

 

u r



F Δ ⁽⁹⁾

The different directional coefficients can be included in a matrix A(t) which depends on the position of the mill, the lead angle and the relation between the cutting coefficients [Altintas 2012; Munoa 2005]. Moreover, instead of using b, axial depth of cut ap will be used because it is usually the user data.

  ^{t  K a} _ ^t 

p

  ^{t }

p

 ^{t  }  

tc

p

A r r

F ( )

sin

p

c, (10)

If it is developed in frequency domain,

 

i



^1ηQ_p^TF_c,p

 

i



Q_q^TF_act,q

 

i



Φ ⁽¹¹⁾

where the frequency response function (FRF) of the system Φ is diagonal and defined as

 

i











²M'K'i



C'



^1

Φ ⁽¹²⁾

and the cutting force in frequency domain is defined as

  ^{A Q}   ^η

F

^

 

^p ₀ ^{i c}

c,

1

i  K sin a  e

^

p tc

p (13)

When Direct Velocity Feedback (DVF) is employed, the control forces appear as viscous damping and the stability lobes increase their limit depth of cut. Such strategy is based on the measurement of vibration velocity and its negative feedback, as shown in (6).

   

  G Q η

F

r G F

















m q

m

q

t t



 i i : DVF

, act

act,



(14)

 



q m



p c,p

T 2 T

i C Q G Q η Q F

K

M    



 

⁽¹⁵⁾

where G=ga∙gv, being ga and gv the actuator gain and the gain matrix introduced by the user, respectively. In a general case, the term related to the actuator gain (QqT

GQm), will have cross values and, hence, all the force will not be introduced directly as viscous damping of each mode. However, if the active damping term is a linear combination of the mass and stiffness matrices, the equation (15) can be decoupled.

K M

G      

⁽¹⁶⁾

 

  

²

m

i

 k

i

 i  c

i

 g

i

 

i

 P

i (17) These equations show, that in the most general case, a model of the system is required in order to diagonalize the active damping term.

However, in most of cases it can be considered that the frequency range of interest is dominated by a single mode i. Therefore, the actuator force term can be simplified as a scalar and it can be considered as proportional force

 







































i m z i m y i m x

vz vy vx a i

q z i

q y i

q x m q

g g g g

, , , ,

, , T

0 0

0 0

0 0











 Q G Q

m z q z vz a m y q y vy a m x q x vx a

i g g g g g g

g   _,_,   _, _,  _,_,

 ⁽¹⁸⁾

When the active system is non-collocated, i.e. the m and q are not the same point, some instabilities can appear depending on the gain sign. It occurs when the signs of the modal vector values are different in both points.

However, in general collocated systems (are usually recommended [Preumont 2002] and then, the equation can be simplified:



, ²



2 , 2 , T

q z vz q y vy q x vx a i m

q GQ g g g  g  g 

Q ⁽¹⁹⁾

If the same gain is introduced for the three directions (gvx=gvy=gvz) the gain is proportional to the modulus of the modal displacement. Moreover, the higher is the modal displacement on the actuation point, the higher is the introduced viscous damping.

Currently, tridimensional actuator has not been developed in the literature, so the actuation in the three Cartesian directions is not possible when only one actuator is employed. When a biaxial actuator [Munoa 2013] is used, all the force capability can be exploited if the modal displacement direction is mainly over a plane, as occurs usually in ram type machine tools. The single axis actuator can be also employed in these cases and if the actuation direction matches with the mode direction, the force can be also exploited totally. If these directions do not coincide, the actuator can also work properly, but some force will be wasted.

Nevertheless, the complexity is highly increased in cases where modal interaction occurs in the frequency range of interest and a complete model of the system is required, since cross terms appear and the sign and values of the gains have to be deeply studied.

The same problems appear with the other control strategies. Direct Acceleration Feedback (DAF) is similar to DVF, but the acceleration signal is fed back instead of the speed. As equation (22) shows, a proportional gain with DAF control law modifies the modal mass and, therefore, the natural frequency can be increased or reduced, depending on the gain sign. A similar effect is caused by Direct Position Feedback (DPF) algorithm, due to the modification of stiffness (see equation (25)).

   

 

G Q ^η

F

r G F















m q

m

q t t

2 act,

act,

i : DAF







 (20)

 



q m



p c,p

T 2 T

i Cη Q F

K Q G Q

M    



 

⁽²¹⁾

 

  

²

m

i

 g

i

 k

i

 i  c

i

 

i

 P

i (22)

   

  G Q η

F

r G F















m q

m

q

t t

 i : DPF

act,

act, (23)

 



q m



p c,p

T 2 T

i Cη Q F

Q G Q K

M    



 

⁽²⁴⁾

 

  

²

m

i

 k

i

 g

i

 i  c

i

 

i

 P

i (25)

The presented control laws are focused on the modification of dynamic parameters. However, the target is changed when the novel Delayed Position Feedback (DelPF) control strategy is employed. The objective of such control law is the distortion of the regenerative effect, by the reduction of the engagement between successive waves. It is based on the feedback of the previous tooth vibration, which is obtained by the introduction of tooth passing period delay feedback. The mathematical formulation is very simple for orthogonal cutting processes [Munoa 2013]. However, the formulation is complicated for milling operations, since the directional matrix has to be considered when the commanded force is calculated.

The force commanded by DelPF can be considered as

     

  G Q η

F

Q η G r

G F











i c

act, act,

i : DelPF























e

t t

t

m q

m m

q (26)

In order to see the influence of the direction matrix on the gain of the new control strategy, the effect is analyzed in frequency domain formulation.

 

^{Q GQ η}

Q A

η Q 



 



  

 K^tca _p^T _p e^_ e^_{ q}^T _m

 ^{c c}

i i 0

p 1

sin

1

Φ- ⁽²⁷⁾

The force direction is very important in this control law since it is focused on the regenerative part distortion and, hence, the gain introduction is not as simple as in the other strategies. The directional matrix has to be taken into account inside the gain, while the same gain gv should be introduced for all Cartesian directions.

T 0 T p

sin

^{q p p m}

tc v a

a g K

g Q Q A Q Q

G ^ 

⁽²⁸⁾

Then,

 

  ^η

Q A

η K

^tc

a Q

_p^T _p

e g

_a

g

_v





 1

^

1

sin

i c

0

p 



1

Φ

- ⁽²⁹⁾

Therefore, the perfect behavior of the control law could avoid chatter for any cutting process when the regenerative part is completely removed (gagv=1).

Moreover, since the acceleration and position are counter phase signals, the acceleration signal can be directly employed in cases where the vibration measurement is performed by an accelerometer. It is named as Delayed Acceleration Feedback (DelAF).

In this case, as equation (13) shows, the feedback has to be negative and the gain should be much smaller.

 

 

^η

Q A

η K^tca Q_p^T _p e ₂g_ag_v

c i

0

p 1 1

sin

c 





 



 ^

1

Φ- ⁽³⁰⁾

Therefore, it is clear that the directional matrix have to be multiplied when the gain is introduced on the controller, in order to reduce properly the regenerative effect. In conclusion, the actuator force is calculated as follows:

  





  

 K a t

g g

t _{a v} ^tc _{q p}^T _{p m}^T

q Q Q AQ Q r

F_act, ^p ₀

sin

(31)

where A0 only depends on geometrical parameters:



K K Z



f _rc, _ac, _s, _e,

0

 

A ⁽³²⁾

Tab. 1: Dynamic parameters of the machine when Y=760mm and Z=550mm.

If the equation (31) is analyzed, it can be observed that when the actuator or measurement does not coincide with the cutting point, it is necessary the knowledge of their modal displacement relation. It means that a model of the system would be required. However, when these points are very close (p≈q≈m), it can be assumed that the modal matrices are the same and then, identity matrix is the result of their multiplication. Moreover, the terms, Ktc, ap

and κ, can be considered constant, so the gain only would depend on the relation of directions and values of the directional matrix terms (33). The same mathematical development can be done for the DelAF.

 

t g_ag_v 



t



q A r

F_{act, 0} ⁽³³⁾

However, when the cutting case is simplified for cases where the chatter is dominated by a single mode, the term QpT

A0Qp becomes into a scalar B0 and the gain is simplified as a scalar, as done in equation (18). In this case, the directional matrix would not have a decisive role.

 



^{e g}



^η

a B

ηK^tc 1 ^ 1 _i  Φ sin ^p ₀ ^ic



-1 (34)

Nevertheless, a bidirectional actuator is recommended in order to improve properly the stability of a machine where the principal modes are defined in a plane. Moreover the proposed formulation can be inaccurate when interrupted cutting is performed, since zero order approximation is used. Therefore, a deeper study of mathematical formulation would be required for these cases.

3 EXPERIMENTAL SETUP

This section presents the machine and the actuator employed for simulations and the subsequent experimental verification. A prototype of a DANOBAT ram type machining center (Fig. 4) was selected, since its stability was previously studied in frequency domain with excellent results [Munoa 2005].

Fig. 4: DANOBAT ram type machining center prototype.

The machine offers a variation of the ram overhang (Z) of 630 mm, height travel (Y) of 1100 mm and 1000 mm of X travel. In the present work, the ram overhang of 550 mm and the height of 760 mm remain constant in order to compare the results of different control laws.

Tab 2. describes the cutting tool, the cutting conditions and the cutting coefficients used in all simulations and tests. Face milling operations with different spindle speeds were tested over the XY vertical plane, being X the feed direction.

Tab. 2: Description of cutting parameters.

Cutting tool

Reference SANDVIK R245-125Q40-12M

Diameter 125 mm

Number of teeth 8

Inserts reference SANDVIK R245-12 T3 M-PM 4030

Lead angle 45º

Cutting conditions

Feed per tooth 0.2 mm/z

Radial

engagement 100 mm

Workpiece

material F1140 Steel

Cutting coefficients

Specific tangential, Ktc 1883 N/mm²

Specific radial, K_rc 0.38

Specific axial, K_ac -0.25

A complete modal analysis was performed in order to obtain the dynamic parameters of the machine, which are summarized in Tab. 1. It can be observed that the flexibility of the machine is mainly defined by XY directions, so the introduction of the control parameters is complicated.

A single degree of freedom (DOF) small electromagnetic actuator [Mancisidor 2013] has been located on the ram frontend (point q). Such device is characterized by its linearity (see Fig. 5) and provides a maximum force of 150 N. The ELMO TUBA servo amplifier has been used in order to convert the voltage commanded by the controller into current. A dSPACE DS-1005 board controller has been employed in order to implement the control laws, while the vibration signal is measured by a piezoelectric accelerometer.

Natural frequency (ωn)

Damping ratio (ξ)

Modal stiffness (k)

Modal vector matrix [Q]

Tool (p) Actuator (q)

33.00 Hz 3.2 % 129.N/μm (0.18, -0.57, -0.80) (0.09, -0.58, -0.95)

34.25 Hz 4.8 % 126 N/μm (0.21, 0.75, 0.63) (0.01, 0.65, 0.76)

50.48 Hz 2.1 % 110 N/μm (0.78, 0.62, -0.06) (0.70, 0.64, 0.00)

53.56 Hz 4.5 % 65.7 N/μm (-0.13, -0.97, 0.18) (-0.11, -0.78, 0.10)

57.25 Hz 2.9 % 15.5 N/μm (0.99, -0.14, 0.02) (0.75, -0.01, 0.03)

63.73 Hz 3.7 % 517 N/μm (-0.80, -0.30, -0.52) (-0.28, -0.38, -0.63)

67.70 Hz 3.1 % 1635 N/μm (0.16, -0.98, 0.07) (0.20, -0.74, -0.12)

78.25 Hz 2.9 % 328 N/μm (-0.07, 0.95, 0.30) (-0.24, 0.74, 0.53)

200.0 Hz 10 % 40 N/μm (0.00, 0.92, -0.40) (0.00, 0.25, -0.10)

Fig. 4: DANOBAT ram type machining center prototype.

Fig. 5: F/V ratio response of the selected actuator.

4 MECHATRONIC MODEL SIMULATIONS The comparison of these four control strategies for orthogonal cutting process was reported in the literature [Munoa 2013], where DVF was showed as the best strategy at almost all spindle speeds. However, the complexity of milling process can distort the good behavior of the presented control laws, since than more than one mode appears and their directions do not match usually with the actuation direction [Munoa 2009].

In general, single degree of freedom devices have been proposed in the literature. However, ram type machine tools present their flexibility on the two bending directions of the ram (see Tab. 1) and then, a biaxial inertial actuator can obtain better results [Munoa 2013]. In this work, both a single direction actuator and a biaxial actuator have been simulated in the cutting process defined in Tab. 2.

In the first case, the actuator described in the previous section has been considered. In this way, the mechatronic model has been employed in order to predict the stability improvement offered by each control law.

The results shown in Fig. 6 present the DVF control strategy as the best one in almost all range of spindle speeds of the stability lobes. It is only overcome in speeds around the sweet spot (maximum stability zone), where DVF cannot improve the stability.

However, as explained before, the machine flexibility is mainly defined by XY directions and more than one mode affects in the critical frequency range. Such issue does not affect negatively in strategies which are focused on the structural response modification. However, when DelPF or DelAF is employed, it affects drastically the results, since, at least, two directions should be controlled in order to reduce properly the regenerative effect, considering the directional matrix.

Fig. 6: Comparison of mechatronic model results of different control laws employed by a single axis actuator.

Fig. 7: Comparison of mechatronic model results of different control laws employed by a biaxial actuator.

In order to verify this statement, a second simulation where a biaxial actuator is simulated has been carried out.

The gain has been the same for both controllable directions and in this way, the real capability of each control law can be observed and compared in Fig. 7.

It can be observed that DVF strategy remains the best in almost all zones of the stability lobes. However, the stability is improved considerably by the DelAF, principally in spindle speeds around the sweet spot and, hence, the maximum productivity (5 mm) is provided by this novel strategy. Therefore, it is proved that DelAF control law requires a biaxial actuator and the consideration of the directional matrix, since if not it cannot obtain a sizeable improvement.

5 EXPERIMENTAL RESULTS

Finally, the proposed control laws have been also compared experimentally in the DANOBAT machining centre prototype. The actuator proposed in section 3 is employed, so the fact that it is a single axis actuator has to be taken into account when the results are compared.

Tab 3. shows numerically the results obtained for the selected position of the ram in the workspace (Y=760mm;

Z=550mm). They can also be observed graphically in Fig.

8. The results are similar to those obtained by the mechatronic model with a single axis actuator. It can be clearly observed that almost all the other zones are clearly dominated by DVF and it is only overcome around the sweet spot. In such zone, DVF cannot improve the stability and then, DelAF achieve the best results.

However, it should be commented that the DelAF results are poor, since a single axis actuator has been tested and hence, the directional matrix could not be considered properly. Therefore the theoretical results obtained for the biaxial actuator should be verified experimentally, in order to compare properly the control laws.

Fig. 8: Comparison of experimental results of different control strategies.

20 40 60 80

0 200 400 600 800

Frequency (Hz)

Magnitude (N/V)

20 40 60 80

-400 -200 0 200

Phase (º)

Frequency (Hz)

200 300 400 500 600 700 800

0 1 2 3 4 5

Spindle Speed (rpm)

Depth of Cut (mm)

Original DAF DVF DPF DelAF

200 300 400 500 600 700 800

0 1 2 3 4 5

Spindle Speed (rpm)

Depth of Cut (mm)

Original DAF DVF DPF DelAF

300 350 400 450 500 550 600 650 700

0 0.5 1 1.5 2 2.5 3 3.5 4

Spindle Speed (rpm)

Depth of Cut (mm)

Original DAF DVF DPF DelAF

Tab. 3: The minimum depth of cut obtained for each control strategy and spindle speed when Y=760mm and Z=550mm.

Fig. 9: Cutting tests (N=610rpm, ap=1.25mm) with DVF control strategy; a) Time domain acceleration signal of machine vibrations; b) surface finishing.

The effect of the DVF control strategy in machine vibrations is clear if the actuator is switched on during the cutting process with chatter problems (see Fig. 9). In the one hand, the vibration acceleration signal is reduced more than a half when the actuator is working. On the other hand, the surface quality is considerably improved, which proves that the control suppress the chatter vibrations.

6 CONCLUSIONS

This work has presented a comparison of the most used model free feedback control strategies for suppressing chatter vibrations, in order to analyse the best control law for a certain milling process. A brief mathematical development of each control strategy is completed in the first part, where the complexity of the laws when they are used in milling operations is shown.

The capital influence of the directional matrix consideration has been demonstrated when Delayed Position/Acceleration Feedback is employed in milling. If such matrix is not taken into account and the flexibility of the machine is defined by more than one mode, the delayed feedback strategies do not introduce the force correctly and in some cases, they can worsen the stability of the system.

Then, the control laws have been tested theoretically by means of a validated mechatronic model, where all strategies can improve the stability of the system. The

simulations of a single axis and a biaxial actuator have been carried out.

The results show that DVF is the best control law for both actuators in minimum stability zone, although the results obtained by DelAF with biaxial actuator are also excellent, principally in spindle speeds close to the maximum stability zone. In this way, the maximum productivity is provided by the novel strategy, when a biaxial actuator is employed and the directional matrix is considered.

Finally, these conclusions have been confirmed by experimental cutting tests with a single axis actuator, where the results obtained have been very similar to those obtained by the mechatronic model. It can be observed that DVF control strategy is the most suitable in most cases. However, the experimental tests with a biaxial actuator would be required in order to obtain the real difference between DVF and DelAF.

7 ACKNOWLEDGMENTS

This work has been partially supported by the PAINT ETORGAI (ER-2012/00019) project and REMEC project (IG-2013/0001136), both funded by the Basque Government.

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