Self-Excited Vibration Drilling Models and Experiments

Self-Excited Vibration Drilling Models and Experiments S. Tichkiewitch (I), G. Moraru', D. Brun-Picard', A. Gouskov3

(1) 3s Laboratory of National Polytechnic Institute of Grenoble, France 2 LSlS - IMS, ENSAM, Aix-en-Provence, France 3 BAUMAN State Technical University, Moscow, Russia

Abstract A nonlinear dynamical model of vibration drilling is presented. It takes in consideration cutting interruption through surface generation equations. The linear stability analysis yields stability charts and the nature of Hopf bifurcation is discussed at critical values of cutting parameters. Dimensionless equations have been employed in order to obtain graphical charts that completely describe the dynamics of a pair of vibrationdrilling head - workpiece material. The analysis of "finite amplitude instability" phenomenon is carried out in time domain by computer simulations. A dynamic cutting fixture was used to run vibration drilling experiments. Based upon simulations and general vibration cutting model described here, the dispersion of the results from experimental work was explained. Important conclusions are drawn concerning forthcoming experiments in vibration drilling. Keywords: Vibration cutting, Modeling, Drilling

1

INTRODUCTION

Vibration cutting tries to take advantage of controlled vibrations instead of eliminating them. It consists in usual machining kinematics upon which additional oscillations (of tool or/and workpiece) are superimposed. Changes brought about by deliberate vibratory cutting consist in decreased average forces and temperatures [I, 21, periodic rest of cutting edges due to disengagement from the workpiece material, increased cutting fluid application efficiency. Low-frequency vibration cutting is well adapted for deep drilling because vibration-assisted chip breakage allows an easier chip removal using coolant supply systems. The technique of self-excited vibration drilling was proposed in [3].A self-vibrating drilling head (SVDH) excites low-energy chatter vibration for certain process parameters by using a low-rigidity part (spring) located between the machine structure and the tool or the workpiece. This low-rigidity part creates conditions for controlled axial regenerative vibrations. The figure 1 shows the principle of the SVDH and proposes a physical model for vibration cutting. The regeneration effect is the result of the dynamic interaction between the mechanical structure (machine and SVDH fixture) and the cutting process (figure 1 (b)): the cut produced at time t leaves small "waves" on the surface that are regenerated with each subsequent pass of the tool. The delay between two consecutive cuts is the central idea of the regenerative effect used to explain chatter phenomenon. The dynamics of the SVDH is given by its mass, damping and stiffness (m, c, k). The spindle (workpiece) dynamics is modeled by the same set of parameters (mw, cw, kw). The origin for feed motion and workpiece vibration motion is given by the initial workpiece position, in order to simplify mathematical relationships. Mathematical models of cutting process dynamics appearing in most of previous works use linear or nonlinear models of interactions between machine and tool dynamics and cutting process itself, without taking into account phenomena occurring in full developed chatter. Generally, the idea was to obtain stability charts

in order to control or avoid chatter. In this paper we discuss the possibility of using axial chatter in order to break chips in deep drilling of hard-to-cut materials. Therefore, one has to consider the cutting interruption as the main reason for amplitude limitation in full-developed chatter (as well as nonlinear cutting force models and machine dynamics), studied for the first time in [4]. The mathematical model of surface generation uses a "memory" variable A (depth of the previous generated surface under the tool edge) and a max function in order to describe cutting interruption. A classical relationship is used for the cutting force, neglecting the influence of the damping of cutting process and its nonlinearities. It is assumed that the cutting interruption is the principal nonlinear effect. The dynamics of the machine is described by a second order differential

x,

SVDH

Workpiece

SURFACE GENERATION instant tool motion

position

+I-

I

CUTTING lNTERRUPTloN

u

MACHINE AND SVDH DYNAMICS

FORCE MODEL force

(b) Figure 1: Self-vibration drilling (SVDH principle and process modeling).

equation (due to the low rigidity of the SVDH compared with the rigidity of the machine-tool slideway system). Differential equations with lag argument have been widely used as models for regenerative chatter. In [5] a dimensionless form of the dynamical system that describes the vibration-drilling process was presented. All coordinate positions were divided by feedrate per cutting edge and time was divided by the delay between subsequent cuts. A more general form (1) of this mathematical model could be obtained by accounting for machine spindle dynamics. Spindle dynamics changes the overall dynamic behavior of the system if chatter frequency is close to one of the spindle natural frequencies. General form of vibration drilling model is:

S(T
a(2)= max(x(2)+ x, (2)- x, (T),a(2 - 1)) X(2) + X,(2) + 4x5px(z) + 4x2p2x(2)= 4 =--x2p2K (6(2)-6(2-1))4

account a rigid spindle system and a constant speed feed motion. This results in:

a(2 < o)= 0 a(2)= maX(X(Z)+ X(2)

a(2- 1))

2,

+ 4x5px(z)+ 4 x 2 p 2x(2) = 4

= --x2pZK

(6(2)-6(2-1))4

4 The dynamic system of vibration drilling is naturally referred to as a class of dynamic system with lag argument [8]. The characteristic feature of these systems is large delay, equal to several periods of free vibration of the tool. Time domain simulations, based on numerical integration of (2),show chip breakage due to cutting interruption for process parameters that exhibits chatter (figure 2). Amplitude limitation by this strong nonlinear phenomenon (cutting interruption) leads to very complex dynamics [9].

4 2, (2)+ 4x5,

p, x, (2)+ 4x2p; x, (2) = (a(2)-6(2-1))4

4 =-X2/3;K, 4

where 6 = A / fz is the dimensionless "memory" variable used in surface generation equations; x = X / fz dimensionless vibration motion of SVDH (tool); X a = Xa / fz dimensionless feed motion; x, = X, / fz - dimensionless vibration motion of spindle (workpiece); z = t / r dimensionless time; r = 60 / N / z - time delay between subsequent cuts; N - rotational speed between tool and workpiece; v = N / 60 - rotational frequency; f - feedrate; z - number of cutting edges; fz = f / z - feedrate per cutting edge; c, k, m - damping, stiffness and mass of SVDH system; c,, k,, m, - damping, stiffness and mass of spindle system; F = K f .DPmrnl.f,:,] - drilling thrust; Kf, p and g - coefficients and exponents of the cutting

law; D - drill diameter; t=c/(2&)

- SVDH damping

- ratio between cutting stiffness ratio; K=(q.F/f,)/k in stationary conditions and SVDH stiffness;

p = v o / v z= f i / ( 2 n v . z ) - SVDH natural frequency divided by cutting frequency;

5,

damping ratio; K, = q . F / f,/kw

= c , / ( 2 6 ) - spindle

- ratio between cutting

stiffness in stationary conditions and spindle stiffness;

p,

= v,/v,

= ,/=/(2nv.z)

- ratio between spindle

natural frequency and cutting frequency Some remarks could be made regarding this general model of vibration drilling. First, these relationships are similar to those obtained for dynamic models of turning or other single point cutting process [6, 71. Second, machine dynamics given by spindle parameters c,, k,, m, could be assimilated for workpiece or complete spindle-workpiece dynamics. Finally, the feed motion is given by the X a function which, in general case, could include some control or command laws imposed by process kinematics. For a constant speed feed motion xa(d = 2. Previously practiced vibration cutting technologies [ I ] used rigid actuators providing axial harmonic excitation (mechanical or hydraulic mechanisms). W e could use the feed control function xa(d to account for these rigid excitation laws by adding a corresponding excitation term. A simplified mathematical model is obtained by taking into

T i m e (s)

Figure 2: Time domain simulations of vibration drilling. The goal of the study of vibration drilling dynamics is to provide information that would permit the control of process parameters in order to create convenient cutting conditions. The criteria used to evaluate the quality of vibration cutting conditions are: dimensionless amplitude of vibrations (restrained to 0.5 - 5 in order to protect the tool), the number of oscillations per cut (recommended limits: 2 - 6) and the cutting rate (the effective cutting time over an oscillation period; it has to be as close as possible to 1). One can see that the number of meaningful process parameters is 4: g, K , /3, 5. Two processes with the same dimensionless numbers are similar in terms of relative parameters of vibration and defined criteria. The dimensionless approach could also be used for experiment planning. Three parameters ( K , /3, Q instead of seven {N, f, m, k, c, D, z} could be used to obtain information on system behavior from simulations and/or experiments. 2 LINEAR STABILITY ANALYSIS Vibration starts in continuous cutting conditions (small amplitude vibration). For continuous cutting conditions, one can eliminate surface generation equations. By considering a linear form of cutting force model the system (2) becomes: X(2)

+ 4x5px(2) + 4x2p2x(2)= =-4x2p2K(1+X(T)-X(Z-l))

(3)

The constant term provide a stationary solution: x(d = - K (the recoil of the tool under stationary cutting force). The

Figure 3: Stability lobes and bifurcation parameters stability of this solution is given by the characteristic equation of the homogenous differential equation obtained from (3): sz +

4 ~ 5 ps .+ 4 ~ ’ p= -4~’P’k-(l ~ - e-‘)

(4)

The analysis of this complex transcendental characteristic equation leads to a “Christmas-Tree’’ stability chart presented in figure 3 [4, 6, 71 (sfabilitylobes).

3 ANALYSIS OF FINITE AMPLITUDE INSTABILITY The linear analysis cannot be used to investigate the phenomena that occur in instability regions. The amplitude of vibration is limited by cutting interruption and the vibration frequency evolves towards a stable limit cycle, depending on process parameters. Moreover, jump phenomena or bifurcation nature could not be explored by linear analysis. In [6] a third degree polynomial form of cutting force relationship [9] was used in order to carry out the closed form calculation and to discuss the nature of Hopf bifurcation with respect to K parameter. It was found that the system exhibits a subcritical behavior for some reasonable suppositions. However, in [7] it was established that by a velocity cubed feedback control one could force the system to behave in a supercritical manner. This altering of bifurcation nature is used in controlled systems, which is not the case in our approach. A subcritical behavior near a bifurcation point means that unstable periodic motion exists around the stable stationary cutting for values of somewhat smaller than the critical value G. This unstable limit cycle defines the domain of attraction of the stable stationary cutting and of the limit cycle which takes place in regions of instability. Time domain simulations of vibration cutting model (2) proved the same subcritical behavior for interrupted cutting [ l o ] (figure 4). However, there is no way to predict, at this stage, by analytical means, the vibration amplitude and frequencies for full developed chatter if we consider cutti ng interruption . The system behavior around the bifurcation point has been studied and experimentally investigated in milling [9] and turning [7] (finite amplitude chatter) and has been mathematically explained in [6].

Figure 5: Cutting rate and number of oscillations per cut. Vibration cutting tries to take advantage of vibrations in cutting processes, therefore a detailed study of system dynamics in instability regions has to be carried out. Numerical simulations were used to investigate the dynamics of vibration cutting. The vibration characteristics (amplitude, frequency) as well as cutting rate and maximal uncut chip thickness were automatically computed over regions in the { K , p) plane. Figure 5 shows simulation results for { = 0.5 and g = 0.8. The instability of limit cycles at the border between two frequency regions was confirmed by simulations. Figure 6 shows the changing of the stable vibration motion (depending on the value of perturbation, another limit cycle takes place in the phase space) for a numeric experiment located at the limit between two regions.

K Figure 4: Bifurcation diagram: subcritical behavior.

Figure 6: Limit cycle changing in time domain.

The existence of several limit cycles leads to the consideration of their corresponding attraction basins. In other words, the question is how much has the system to be perturbed in order to change the limit cycle, and, consequently, the amplitude and the vibration frequency. This sensitivity could also explain the dispersion of experimental results discussed in the next section Simulations shows that in regions faraway from stability limit (where previous vibration drilling experiments were carried out) possible chaotic phenomena could arise (figure 6). Therefore, a small modification in initial conditions could change vibration characteristics. 4 EXPERIMENTS Experiments were carried out in order to validate the models of vibration drilling. A "gun-drilling'' machine and a SVDH were used to drill holes of 0 5 mm in 35NCD6p steel (Rm = 880/1030 N/mm2, Rp0.2 t 700 N/mm2, A% t 12, KCU(L) t 80 J/cm2). Different cutting speed (2000 to 5100 rpm), feedrate (7.5 to 20 pmhev) and SVDH stiffness (160 to 900 N/mm) were used. Frequently, speeds from 1200 to 1500 rpm and small feedrates (smaller than 15 pmhev) are used to obtain convenient cutting conditions in deep drilling of this steel. Experiments produced vibration cutting conditions, as predicted by theory but a dispersion of results (in terms of frequencies and amplitudes of vibration) was reported. For much of experiments we found an amplitude factor between 3 and 40, and a number of oscillations per cut too big (up to 10). These are the reasons for the low tool life observed.

(a) continuous cutting

(b) high cutting rate

(c) low cutting rate

Figure 9: Vibration drilling experiment planning Vibration cutting quality parameters depend on regions in the { K , p} plane where the experiments are carried out. The experiments were situated close to the possible chaotic regions revealed by simulations (figure 9). This is most likely why a dispersion of the results was reported. Chaotic phenomena could be avoided by choosing adequate drilling parameters and SVDH characteristics in order to bring the experiment in { K , p) plane close to the stability limit (figure 9). 5

CONCLUSIONS

Vibration drilling is a solution for drilling deep holes with high productivity, suppressing retreat cycles. Based on the use of axial chatter, this technique decreases average forces and temperatures and allows an easier chip removal (particularly important for small diameters). The numerical simulations and the experiments carried out have validated mathematical models of vibration drilling. The complex nonlinear dynamics has been investigated and an improved experiment planning was suggested, based on dimensionless parameters and stability analysis.

Figure 7: Comparison of chip morphology Figure 7 illustrates three typical chip morphologies: (a) conventional drilling with continuous chip; (b) vibration drilling with a small number of oscillations per cut and high cutting rate; (c) vibration drilling with an excessive number of oscillations and insufficient cutting rate. In figure 8 two results of vibration drilling experiments are presented, located in { K , p) plane in figure 6. The first vibration experiment produced periodic, almost harmonic oscillations. The second experiment, provided by cutting conditions situated faraway from stability limit in { K , p) plane, reveals a possible chaotic behavior (figure 8 (b)). I

(a) working point 1 (figure 6) - periodic vibratory motion

(b) working point 2 (figure 6) - possible chaotic motion Figure 8: Tool vibration in vibration drilling experiments

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