2011 chatter vibrationsurveillancebytheoptimal-linear

7/25/2019 2011 Chatter Vibrationsurveillancebytheoptimal-linear

1/17

Chatter vibration surveillance by the optimal-linearspindle speed control

Krzysztof J. Kalinski 1, Marek A. Galewski n

Gdansk University of Technology, Faculty of Mechanical Engineering, ul. G. Narutowicza 11/12, 80-233 Gdansk, Poland

a r t i c l e i n f o

Article history:

Received 29 June 2009

Received in revised form

29 August 2010

Accepted 2 September 2010Available online 15 September 2010

Keywords:

Chatter vibration

High speed milling

Optimal control

Variable spindle speed

a b s t r a c t

The paper concerns self-excited chatter vibration during high speed slender ball-end

milling. Non-stationary cutting process, with inclusion of various approaches towards

dynamic characteristics of the process, is described. Dynamic analysis of the milling process

is performed and dynamics of controlled closed loop system with time-delay is presented.

In order to reduce vibration level, instantaneous change in the spindle speed appears as a

control command, and thusthe method of vibration surveillance by the spindle speed

optimal-linear control is developed. Presented cutting models have been applied for the

proposed method and procedure of the chatter vibration surveillance with a use of variable

spindle speed has been developed. Computer simulations are performed for selected cases

of ball-end milling at constant and variable spindle speed. The results of them are

successfully confirmed by experimental investigations on the Alcera Gambin 120CR milling

machine equipped with the S2M high speed electrospindle.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Contour slender ball-end milling is a very frequent case of operations performed on contemporary production centres.

Toolworkpiece relative vibration plays principal role during the cutting process. Due to existence of certain conditions, it

may lead to a loss of stability and cause generation of self-excited chatter vibration [1]. There are many different methods

for reduction and surveillance of the chatter vibration. However the methods represent, from the point of view of possible

implementations, a lot of disadvantages. The latter concern the following:

vibrating cutting[2]. It is rather impracticable for milling operations; control of instantaneous toolworkpiece relative position[3,4]. Successful application of it needs an interference with

the machine tool structure. Thus, it is inconvenient in practice; active damping of a tool or a workpiece[57]. Suitable actuators (e.g. piezoelectric or electromagnetic), as components

of on-line real-time control systems, generate the applied dynamic forces. Although the methods have great potential

for possible applications, efficiency of the surveillance has to be doubtful. The difficulties lie in obtaining sufficient

sampling frequency of the real command forces;

matching the spindle speed to selected dynamic properties of the system [810]. Although this way seems to be soeasily applicable, the success is limited to only short-time operations when chatter vibration has not been developed

still; and

Contents lists available atScienceDirect

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Mechanical Systems and Signal Processing

0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ymssp.2010.09.005

n Corresponding author. Tel.: +48 58 347 11 21.

E-mail addresses: [email protected] (K.J. Kalinski),[email protected] (M.A. Galewski).1 Tel.: +48 58 347 14 96.

Mechanical Systems and Signal Processing 25 (2011) 383399
http://-/?-http://www.elsevier.com/locate/jnlabr/ymssphttp://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.ymssp.2010.09.005mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.ymssp.2010.09.005http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.ymssp.2010.09.005mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.ymssp.2010.09.005http://www.elsevier.com/locate/jnlabr/ymssphttp://-/?-


2/17

on-line spindle speed optimisation [11]. Generally the approach above produces unsuccessful results for modernmachine tools.

The spindle speed variation appeared to be an attractive class of these methods [1,11]. One of the chatter reduction

methods depends on the programmed spindle speed control[8]. The programme generates step-changing spindle speed at

desired switching times. In practice, their values are 0.080.4 s. However, good efficiency of the surveillance has been

evidenced only in case of the groove flat milling, and the spindle speed values not exceeding 4000 rev/min. In turn, the

spindle speed optimal control [8,12,13] generates the instantaneous spindle speed which has a tendency to approachinfinity.

That is why the problem of toolworkpiece vibration surveillance is still looking for new solutions, especially in scope of

high speed cutting. The latter supports originality and novelty of the method of vibration surveillance by the spindle speed

optimal-linear control, being applied in this paper. Results of experimental investigation of high speed milling on the

Alcera Gambin 120CR milling machine evidenced good efficiency of vibration surveillance. Chatter vibration has been

reduced and variable spindle speed programmes are possible to be performed in reality.

2. Self-excited chatter vibration during the cutting process

Vibration observed in the machineworkpiecetool-holder (MWTH) system may be resulted by various reasons. The

most important opportunity lies in the feedback interaction between the massspringdamping system of the milling

machine and the milling process. The existence of feedback interaction is not sufficient condition to trigger self-excited

vibration, but it makes the vibrating system potentially unstable. In such a kind of the system, under certain conditions, the

feedback interaction may lead to loss of stability and generation of a limit cycle (i.e. caused by periodic loss of contact

between the tool and the workpiece [1,14]). Relative displacements between the tool and the workpiece during cutting

process imply changes in cutting layer dimensions. The changes above trigger changes in values of cutting forces. In turn,

changes in the cutting force cause relative displacements of the tool and the workpiece. In case of self-excited chatter

vibration, a great danger is due to inner interactions in the MWTH system, and the tool path regeneration phenomenon

[1,14]. The other reasons of the chatter generation are usually of a minor importance.

Analysis of dynamics of a slender ball-end milling of the rigid workpiece has been performed, taking into consideration

following assumptions (Fig. 1):

In the milling machine structure, the spindle together with the tool and the table with the workpiece are separated intosubsystems that experience relative motions. It is shown[1,8,15]that those subsystems are essential from the point of

view of cutting dynamics, while contribution of the other components of the machine tool structure into toolworkpiece relative vibration is insignificant.

Only flexibility of the tool is considered. All other components are treated as perfectly rigid. Former experimentalinvestigations of the considered milling centre showed, that there was no vibration having natural frequency below

1500 Hz[12].

Coupling elements (CE) are applied in order to idealise cutting process interaction. They are placed at conventionalcontact points of tool edge and workpiece.

The effect of first pass of the edge along cutting layer is idealised as proportional feedback, and the effect of multiplepasses is idealised additionally as delayed feedback.

The tool rotates at spindle speed n . The workpiece, length Lw, moves itself at feed speed vf.

Cutting layer thickness hl is defined as a distance between two identical positions of successive cutting edges in space

and is measured in direction being perpendicular to instantaneous cutting speed vc. In the milling process, desired cuttinglayer thickness hDlis not constant and depends on instantaneous angular position jl of cutting edge no. l (Fig. 1). Thus, itcan be described by a simplified relationship[8,14]:

hDltffifzcosjlt, 1

wherefzfeed per edge; fz=vf/(nz) and znumber of cutting edges.

While the tool rotates itself, it will experience transverse vibration. The latter implies that instantaneous cutting layer

thickness, for current cutting edge no. l, varies in time about its desired value and the edge leaves a trace on the machined

material. The trace influences the pass of the subsequent cutting edge. These two effects are called: inner and outer

modulation of the cutting layer thickness[1,14]. Taking the above into consideration, instantaneous cutting layer thickness

hl(t) is described, i.e. [8]:

hlt hDltDhlt Dhlttl, 2

where Dhl(.) is the dynamic change in cutting layer thickness andtlthe time-delay between identical position of a currentand a previous cutting edge.

K.J. Kalinski, M.A. Galewski / Mechanical Systems and Signal Processing 25 (2011) 383399384


3/17

In case of milling operations, time-delaytldepends on a number of cutting edges z, as well ason the spindle speed n.Then, it can be expressed as follows [8]:

tl Z jl

jlj0l

60

2p

dj

nj, 3

where j0l is the difference between angular position of cutting edge no. l and cutting edge no. l-1, and n(j) theinstantaneous spindle speed as a function of angular position j.

3. Dynamic characteristics of the cutting process

As a result of discrete modelling of the cutting process, a system composed of flexible finite element (FFE) no. e and

coupling elements (CEs) is created (Fig. 2). The FFE is the EulerBernoulli bar element [16] having length Lt(i.e. active

length of the tool) and fixed at its one end in the tool holder. It also allows us to include influence of the spindle speed on

the tools dynamic performance (so-called gyroscopic effect [17]).

The CEs are placed at instantaneous positions of cutting edges. The instantaneous position of tool edge is described by

anglejl=jl(t) and certainly corresponds to instantaneous position of CE no. l. Theyl1,yl2, andyl3axes are principal axes offeedback interactions of this CE. During the cutting process not all of the edges cut the material at chosen instant of time.

Fig. 1. A scheme of a slender ball-end milling process.

K.J. Kalinski, M.A. Galewski / Mechanical Systems and Signal Processing 25 (2011) 383399 385


4/17

Cutting edges having instantaneous contact with the material are called active, while the remaining ones are called

inactive.

Additionally, following particulars are described on the cutting process scheme (Fig. 1):

rake angleg0 and clearance anglea0, as the items of the edge geometry in orthogonal plane, cutting force component Fyl1 (so-called, principal cutting force) acting along cutting speed vc, cutting force component Fyl2 acting along actual cutting layer thickness hl, cutting depth ap, milling widths B1 and B2; in case of full-immersion ball-end milling B1=B2and Dm= B1+B2, conventional contact point between tool and workpiece S, and local immovable coordinate system xe1, xe2, xe3 of FFE no. e.

Coupling elements allow us to connect geometrical quantities, such as displacement along principal cutting force,

instantaneous cutting layer thickness and instantaneous change in cutting depth, with the force components acting along

relevant directions[1,8,14]. Coupling elements are described by matrices of the feedback interaction, which are, in general

case, non-diagonal. They are mostly defined as appropriate Laplace transfer functions. The latter permits us to consider thedesired force and the time-delayed feedback interaction. In the considered case, an influence of angular displacements on

Fig. 2. Discrete model of the process in which ball-end mill has two cutting edges.

K.J. Kalinski, M.A. Galewski / Mechanical Systems and Signal Processing 25 (2011) 383399386


5/17

cutting forces of CE is neglected. Only longitudinal displacements, as results of the force components acting in orthogonal

plane (i.e. Fyl3=0) (see Sections 3.1 and 3.2), are included.

In order to define cutting forces, a lot of different models had been applied in the past [1,14]. In this paper, two

mechanistic approaches are worth of being emphasised. The first one depends on application of proportional model of the

cutting dynamics[1,8], while the second relates to utilising the NosyrevaMolinari model of the cutting dynamics[18].

3.1. Proportional model of the cutting dynamics

Cutting force in the milling process depends on instantaneous cutting layer thickness (Eq. (2)), average dynamic specific

cutting pressure and cutting depth [1,8,14]. For conventional contact point between the tool edge and the workpiece

(CE no.l,Fig. 2), Cartesian coordinate systemyl1,yl2,yl3is introduced in order to define components of the resultant cutting

force. Finally, after including the effect of cutting edge being out of the material, when vibration approaches to significant

level, several components of instantaneous cutting force are described with a use of following equations[8,12]:

Fyl1t kdlaphDltDhlt Dhlttl forhDltDhlt Dhlttl40,

0 forhDltDhlt Dhlttlr0,

( 4

Fyl2t mlkdlaphDltDhlt Dhlttl for hDltDhlt Dhlttl40,

0 for hDltDhlt Dhlttlr0,

( 5

Fyl3t 0, 6

wherekdl is the average dynamic specific cutting pressure, m l is the cutting force ratio.Force Fyl3=0, because resultant instantaneous cutting force acts in orthogonal plane.

The regeneration phenomenon makes cutting force dependent on the change in cutting layer thickness due to the pass

of previous cutting edge. The latter means that time-delayed feedback interaction really occurs. Closed loop delayed

system can be stable or unstable, in dependence on a current value of the time-delay. In case of milling, this time-delay

depends on difference between angular position of cutting edges and instantaneous spindle speed (3). Consequently, in

case of tools with a constant angle pitch between cutting edges, time-delay can change its value only with the spindle

speed changes. This implies a possibility of the process stability control by such instantaneous changes in the spindle

speed, which allow us to avoid conditions favouring the occurrence of chatter generation.

3.2. The NosyrevaMolinari model of the cutting dynamics

Proportional model is often applied for conventional, lower than the HSM (High Speed Machining) range of cutting

speeds. However, there is a need to achieve simulation results being comparable with those obtained by the experiment.

While cutting speed grows, errors caused by a simplified proportional approach will be accumulated. Therefore, it is

necessary to modify this model considering an influence of the cutting speed on instantaneous cutting forces. Nosyreva

and Molinari proposed such a model for one-dimensional cutting [18]. If we consider the cutting path regeneration, and

define the milling process damping time-constants Tyl1 and Tyl2, whose values remain known functions of the vccutting

speed, suitable Eqs. (4)(6) describing several components of the cutting force will have the modified forms as

follows[12]:

Fyl1t kdlaphDltDhltTyl1D

_h lt Dhlttl forhDltDhltTyl1D_h lt Dhlttl40

0 forhDltDhltTyl1D_h ltDhlttlr0

8

2011 chatter vibrationsurveillancebytheoptimal-linear

Documents