Dominant Vibration Frequencies in Milling Using Semi-discretization Method

Zoltan Dombovari*, Mikel Zatarain, Tamas Insperger

Budapest University of Technology and Economics, Department of Applied Mechanics, Budapest 1521, Hungary

Ideko-Danobat Group, Department of Mechanical Engineering, Elgoibar, Gipuzkoa, dombo@mm.bme.hu

Abstract: Vibration frequencies of high speed milling operations are analyzed. The stability properties are determined by a finite dimensional approximation of the monodromy operator of the system using the semi-discretization method. According to the Floquet theory, loss of stability is associated with multiple vibration frequencies that are given as a base frequency plus an integer multiplier of the tooth passing period (or the characteristic frequency of the cutter). The dominant vibration frequencies that are usually associated as chatter frequencies are however hidden among the infinitely many harmonics. In this paper, it is shown that the amplitudes corresponding to the individual frequency harmonics can be determined in a simple way without increasing the computational cost. The method can be used to determine the dominant frequency components that helps in the identification of the interactions between different modes and the spindle speed. Keywords: high speed machining, regenerative effect, delay differential equation, semi-discretization.

1. INTRODUCTION

High speed machining plays more and more important role in the machining industry due to the demand for increasing productivity. For high spindle speeds, the rotational (or tooth passing) frequency of the milling cutter is in the region of the essential modal frequency of the tool/toolholder/machine structure. As it was pointed out in the pioneering works of [Tlusty, 1954] and [Tobias, 1965], stability properties of the milling operation can essentially be increased in the high speed region. Since then several numerical techniques were developed to determine stability diagrams in the plane of the spindle speed and the depth of cut, like the single-frequency approach by [Altintas et al., 1995], the multi-frequency analysis by [Budak et al., 1998; Merdol et al., 2004], the semi-discretization (SD) method by [Insperger et al., 2002] or the time-finite element analysis by [Bayly et al., 2003]. The single- and the multi-frequency approaches are frequency domain based methods, while the semi-discretization method works in the time domain. An essential feature of the single- and the multi-frequency approaches is that they are able to predict the dominant chatter frequency that helps to

figure out which modal modes of the machine are dangerous with respect to chatter. According to the Floquet theorem the chatter signal typically contains multiple frequency components due to the periodic nature of the process that can be presented in frequency diagrams [Insperger et al., 2003]. The components corresponding to large vibration amplitudes can be considered as the dominant frequencies. In the current paper, we show how the dominant frequencies can be predicted by the SD method.

2. MILLING MODEL

The relative vibration of the tool-tip is described in modal space defined by the modal coordinates q = col( q1, q2, …, qn ) (see figure 1). The modes are supposed to be real, that is, the damping is proportional, consequently, the equations of motion has the following form

))(),(,()(][)(]2[)( T2,n,n jmmm tttttt qqFUqqq , Nj ...,,1 , (1)

where [2 m n, m] and ][ 2,n m are diagonal matrices with m and n, m being the

damping ratio and the natural angular frequency of the mth mode (m = 1, …, n). Due to the regenerative effect, the resultant cutting force F depends on the current q(t ) and N previous positions q(t – j), where N is the number of regenerative delays in the system. Note that F already contains the cutting force contributions of all flutes.

Figure 1; Model of milling process with general modes. This general representation of the milling operation describes multiple delay cases that typically occur for non-uniform pitch angle tool [Budak, 2003] or for serrated tools [Merdol et al., 2004; Dombovari et al., 2010] in contrast with the conventional models with a single delay equal to the tooth passing period. In (1), the force is distributed to the modal directions by the mass normalized modal matrix

)(diag mcPU , where ][ 21 nPPPP ,

Pm is the modeshape vector and 2/1T )( mmmm mc PP is the normalization parameter of the mth mode, where mm is the modal mass corresponding to Pm. Since, the tool is rotating, (1) is time-periodic at period T, which is the tooth passing period or an integer divisor of the rotation period depending on the type of the cutter. The solution can be rewritten as the sum of a stationary and the perturbed motion, thus

)()()( p ttt uqq , )()( pp Ttt qq . (2)

The linear variational system of (1) can be determined in the form

N

jjjmmm tttttt

1

2,n,n )()()())(]([)(]2[)( uHuHuu ,

where the coefficient matrices are

))(),(,()(

)( pp jj

j tttt

t

qq

q

FH ,

N

jj tt

1

)()( HH . (3)

Note that these matrices are also time-periodic at T, that is, H j (t) = H j (t+T ). Consider that, if (1) is linear, the costly calculation of qp by the boundary value problem defined with (2) and (1) is not necessary since (3) becomes independent on the delayed coordinates q(t – j) of the tool. Finally, the dynamic model can be prepared for linear stability analysis in its first order representation [Insperger et al., 2008] as

N

jjj ttttt

1

)()()()()( yRyLy , (4)

where L (t) = L (t+T ) and R j (t) = R j (t+T )’s are the linear and the retarded time-periodic matrices and the new coordinates are defined as

)(

)()(

t

tt

u

uy

.

The stability of (4) can now be analyzed using the semi-discretization (SD) method.

3. STABILITY OF THE VARIATIONAL SYSTEM

According to the theory of DDEs [Hale, 1977, Stepan, 1989], the linearized equations of motion at (4) generates an infinite dimensional function space defined by the shift as

)()( tt yy , where ]0,[ max and ),,,max( 21max N .

Using an arbitrary initial state ( ) the solution can be given by the so-called fundamental operator U ( t, ) in the following way

))(,()( tUt y ,

where the linear operator U ( t, ) gives the connection between the actual state yt ( ) and the initial state ( ). According to the extension of the Floquet theorem to DDEs [Farkas, 1994], the stability of the system is fully determined by the fundamental operator which maps the initial state y0 ( ) = to yT ( ) , the corresponding monodromy operator is denoted by M = U ( T, y0 ( ) ). The solution after one period is given by

))(()( 0 yy MT . (5)

Moreover, the Floquet theorem also claims that a general solution at the present time can be written as a product of a time-periodic and an asymptotic term in the form

tt ett )()()0( ayy , (6)

where a(t) = a(t+T ) and the complex number is called characteristic exponent. In this manner an element of the function space at the period T is given by

ee TT )()( ay . (7)

If one substitutes (7) into (5) an eigenvalue problem can be formulated as

0s ))()(( IM , (8)

where I is an identity operator, the nonzero complex eigenvalue = e T is called

characteristic multiplier and, according to (7), the complex eigenvector has the form

e)()( as . (9)

Note that s ( ) is defined over the interval [– max, 0]. Equation (8) has a nontrivial solution if

ker( ) M I {0}.

The orbital stability of the stationary solution qp of (1) is asymptotically stable if all the infinitely many characteristic multipliers have magnitudes less than one, that is, | l | < 1 (otherwise qp is unstable). Note that, qp is critically stable if one of the multipliers is | c | = 1. The original system (1) goes through a secondary Hopf bifurcation if c is complex with its conjugate pair c . Period doubling (flip) or cyclic

fold (saddle-node) bifurcation occurs if c = – 1 or c = 1, respectively.

4. SEMI-DISCRETIZATION

The SD method is based on the discretization of the infinite dimensional state space and the periodic coefficients according to the Floquet theory [Farkas, 1994]. The zeroth-order SD method was introduced for general DDEs (including distributed and time-varying delays) in [Insperger et al., 2002], while the first- and higher-order SD techniques were presented for point delay in [Elbeyli et al., 2004, Insperger et al., 2008]. Here, the first-order method is applied. The point of the method is that the delayed terms y(t – j) are approximated over the discretization interval t[ti , ti +1] as

)()1(

))1(()(

)(~,

jit

jjjit

jjjij r

ritr

ritt yyy ,

where, =t=T / k with k being an integer approximation parameter and rj = int( j / +1/2). Note that )())1(( 1

jrijittr yy , )()(

jrijittr yy and

ti = i t. This way, DDE (4) is approximated by an ordinary differential equation (ODE) over the time interval t[ti , ti +1] as

N

jjijiji ttt

1,, )(~)()( yRyLy , (10)

where the time-periodic coefficient matrices are given by the averages

1

d)(1 it

it

i ttt

LL and

1

, d)(1 it

it

jij ttt

RR .

With the help of the analytical solution of equation (10), one can formulate a linear map which projects the solution to the next time step

iii zBz 1 , (11)

where zi = col(yi , yi –1, …, yi – r ) with r = int( max / +1/2). The matrix Bi is actually the discrete representation of the solution operator U ( t, ) over the time interval t[ti , ti +1], where zi corresponds to the initial state . Multiple application of (11) results in

iiikikiiki zBBBBzz 121 ... ,

where the transition matrix is a finite dimensional discrete approximation of the infinite dimensional monodromy operator M. The (finitely many) eigenvalues of the transition matrix are close to the multipliers of the monodromy operator depending on the step size of the discretization. This explains why the stability diagrams constructed by the SD method reliably approximates the exact stability boundaries. The discrete representation of the eigenvalue problem (8) can be formulated as

0SI )( , (12)

where I is a unit matrix and S is the discrete approximation of the eigenvector s ( ) defined at (9). If | l | < 1 for all l, then the system is predicted to be stable, otherwise, it is unstable.

5. MULTIPLE VIBRATION FREQUENCIES

The stability boundaries are the lines in the parameter plane where critical multipliers (i.e., | c | = 1) occur. In one side of these boundaries (usually below), the operation is stable and the tool vibration tends to the time-periodic stationary orbit qp. In the other side of the boundaries, the system loses its stability and it approaches a higher amplitude stable attractor as a threshold of the unstable motion. These non-smooth orbits are referred as chatter vibration in the machine tool industry. Note that this ‘outside’ attractor, according to its non-smooth sense, can simply be a stable periodic orbit, multiple-periodic orbit or a stable chaotic attractor [Dombovari et al., 2009]. All of these structures are basically originated from the orbits occurred at the linear stability limit, which means the spectrum of this non-smooth orbits (chatter) contains the mark of the just-bifurcated orbits. This explains the practical observation that the measured spectrum is close to the predicted one using only linear theories. According to (7) and (9) the solution at each period is given

...)()(...)()()( cccccc222111 ssssy llllTl bbbb , (13)

if the initial state is in the form 10 )()( k kkb sy . Equation (13) shows that all of the

terms will die out except the critical one(s) as

)()()( cccccc ssy llTl bb ,

if l is sufficiently large, since all the other characteristic multipliers are in modulus less than one. Consequently, the characteristic exponents in (6) that correspond to the critical characteristic multipliers can be derived using the inverse formula (complex logarithm) introduced after (8), i.e.

Tq qc,cc )π2arg(i||ln ,

where c, q = c+ i c, q (q can be any integer number) and the multiple frequencies can be expressed as

qq bc,c, , where π/T2 . (14)

Here c, b:= c, 0 = (arg c) / T is the base frequency that can be calculated directly from the critical multiplier and it indicates the lowest possible vibration frequency, which satisfies the linear map (5) formulated by the Floquet theorem, i.e., T c, b[– 2, 2]. Equation (14) implies that infinitely many vibration frequencies arise in the spectrum that are separated by the period-frequency as it was shown in [Insperger et al., 2003] and also coincides with [Budak et al., 1998]. Note that along the stability boundaries, the critical asymptotic part c is zero, but in practice during the preparation of a stability chart the calculated points are never exactly lying on the border. Thus, for the further investigations this also needs as c = ln | c | / T. Using the SD method at a point in the vicinity of the stability border, the approximated spectrum of the monodromy operator can be calculated and the critical multiplier c and the discretized version of the corresponding eigenvector Sc be determined numerically (see (12)). The definition of the eigenvectors at (9) gives the possibility to construct the time-periodic term of the critical eigenvector and to write it in Fourier series form

q

qqee i

c,ccbc,)()( Asa , (15)

where c, b = c+ i c, b is one of the critical characteristic exponent. In order to obtain the dominant vibration level, the velocity of the vibration should be considered. The Fourier coefficients of the vibration velocity are given by Vc, q = i q A c, q according to (15). The dominant vibration frequency corresponds to the maximum (highest infinite norm) of the calculated Fourier coefficients. Thus, the dominant frequency ratio qd hidden by the Floquet theorem can be selected as

||||max

c,dc,maxc, qqqVV V .

If max = T, then this step is equivalent to the FFT analysis of the discrete representation of the critical periodic term (15) that, at the same time, gives qd. Consequently, the dominant vibration frequency is given by

dbc,dc, q .

This way, the strength of each vibration frequency can be determined and represented in the frequency diagrams. Note that for an uneven pitch cutter, the condition max = T does not hold. In this case, the size of the domain of the delayed time should be extended over the principal period T, that is, [–, 0]. This increases the dimension of transition matrix, and therefore may increase the computational cost.

6. CASE STUDY

In this section the calculation is shown through an example half-immersion down-milling operation with a three fluted cutter taken from [Altintas et al., 2008]. Two modes in orthogonal directions of the tool/toolholder/spindle structure are considered with the modal parameters shown in table 1.

mode n,m (Hz) m km (N/m) modal matrix x 510 0.04 96.2

10

01P

y 802 0.05 47.5

Table 1; the modal parameters of the modes considered in the calculation.

The calculation were done using first order SD method with NSD=90 elements to discretize the state yt (). The stability chart (see figure 2b) was created by fractal method based on triangle elements. The spectrums of the periodical term of the critical eigenvectors a c () were calculated at the closest points of the stability limits. In the frequency plot in figure 2a the dominant vibration frequencies are indicated by black color along the stability limits. The strengths of the other harmonics were indicated by grayscale no below that the practical limit of 20%. Certain resonant frequencies m = n,m /N of the modes are pointed out in figure 2 referring to the resonant spindle speeds, the flip regions and the mode interaction zones [Muñoa et al., 2009]. In figure 2a one can follow that the dominant vibration frequencies are in most of the cases close to the natural frequencies. Point A corresponds to period doubling (flip) bifurcation that can be recognized in figure 2a as the frequencies are lying on a straight line of slope 1/2. Points B and D are in the mode interaction zones where the harmonics of the dominant vibration frequencies resonates with the other mode. At point D, this effect is complicated with another strong harmonic of the same dominant vibration frequency. Point C represents the simple case, when only the dominant frequency is strong. The multipliers with the unit circle is depicted in figure 2b and the type of the stability lost can be followed in this way at the picked points (A, B, C, D).

Figure 2; a) shows the dominant vibration frequencies (black) along the stability

border. The strengths of the harmonics indicated by grayscale. n,m’s (m =1, 2) are the natural frequencies of the modes considered in the calculation. Panel b) shows the

stability chart and the position of the instance calculations (A, B, C, D). The unit circles with the multipliers also depicted in this panel close to the picked points.

Figure 3 shows the discretized critical eigenvectors s c ()’s (figure 3a) and their periodic terms a c ()’s (figure 3b). The spectrums of a c ()’s and the FFT’s of time domain simulations of the variational system (4) by the standard DDE23 routine (MATLAB) are depicted in figure 3c. One can immediately recognize (figure 3a) that the eigenvectors are not time-periodic as it was expected at (9).

Figure 3; a) shows the critical eigenvectors of the transition matrix. In panel b) the periodic terms of the critical eigenvectors are depicted. c) presents the FFT’s of the

periodic terms (denoted by black dots) and the FFT’s of time domain simulations of the variational system (4) (denoted by continuous line). The triangles mark the base

vibration frequency c,b. Dashed lines denotes the tooth passing frequencies.

They oscillate around zero since those eigenvectors are derived from the variational system where in fact the amplitude of the theoretical stationary solution is zero. This is not true for the periodic terms (e.g. point A in figure 3b), which may have non-oscillating component, too. According to (14), this only means that the dominant vibration frequency is actually the calculated base frequency bc, , that is, qd = 0. In all

of the other cases the periodic terms seem to oscillates around zero, because they have different dominant vibrations frequency as the base one, thus, qd is non zero (see the triangles denoting the base frequencies in figure 3c).

7. CONCLUSIONS

The motivation to this paper was that the calculation of the dominant vibration frequencies of time-periodic milling operations using semi-discretization method is not trivial. It was shown that the drawback of the Floquet theorem, namely that the frequency ratio between the vibration frequency and the forcing frequency is hidden, can be avoided. This frequency ratio (qd) basically is concealed in the eigenvectors of the Floquet transition matrix. It was pointed out that the dominant vibration frequency can be determined using the time-periodic term of the critical eigenvector along the stability limit. Moreover, in this way other strong frequencies can be found and even the mode interaction zones can be indicated. A case study was presented where two modes

are considered in conventional down-milling operation and different kind of vibration patterns of the tool shows up.

ACKOWLEDGEMENT

This work was partially supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and the Hungarian National Science Foundation under grant no. K72911 and K68910 and HAS-BUTE Research Group on Dynamics of Machines and Vehicles.

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