Stability of transversal vibrations in high-speed drilling and the influence on the processing precision.

Popescu, Daniel ; Cherciu, Mirela

1. INTRODUCTION

Gyroscopic systems include as examples translating and rotating strings, beams, membranes and plates. These systems posses combinations of system parameters that produce vanishing eigenvalues. Such combination of system parameters are designated herein as critical system parameters and their vanishing eigenvalues and the corresponding eigenfunctions as critical eigenvalues and critical eigenfunctions.

One interesting phenomenon for gyroscopic dynamic systems is that unstable conservative systems can be stabilized by increasing the gyroscopic forces. Criteria to determine reasonably small gyroscopic forces for stabilization are important for practical problems. Stability criteria for gyroscopic systems have been formulated for both discrete and continuous systems. For discrete systems, the criteria usually relate the definiteness of a combination of the coefficient matrices in the equation of motion to the stability of the system (Huseyin et al., 1983; Huseyin, 1991). The extension of these criteria to continuous can only ensure stability when the system stiffness operator is positive definite (Shieh, 1971; Wickert & Mate, 1990).

Other stability criteria have been developed to examine continuous systems whose operator is not definite, but these criteria usually require knowledge of the system eigenfunctions (Shieh, 1971), which is normally as difficult to obtain as the eigenvalues themselves.

2. THE MATHEMATICAL MODEL

It is considered the general equation of continuous gyroscopic systems (Renshaw, 1996):

Mii + Gu + Ku = f (1)

where: M, G, K--differential linear matrices in spatial domain

u--displacements

f--forcing term

In order to obtain the mathematical model for the main spindle vibrations, the following assumptions are made:

a. there are no distributed superficial forces on the external spindle surface

b. there are no additional fulcrums or joints that can produce shocks

c. the initial state of the spindle is unstrained

d. a plane section normal on the spindle axis before deformation remains plane without keeping perpendicularity

Under these assumptions, for the mathematical model of transversal vibrations given by (1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where: [u.sub.1](x), [u.sub.2](x)--transversal deformation of the spindle on two perpendicular axes; [f.sub.1](x), [f.sub.2](x)--external load; [OMEGA]--spindle angular speed; p(x), c(x)--coefficients depending on section characteristics and spindle material. We have:

p(x) = I(x)/A(x);[c.sup.2](x) = EI(x)/[rho]A(x) (3)

The limit conditions in this case are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In this case larger deformations are allowed and applying the finite Fourier transform gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

If the angular speed of the spindle is smaller than the critical value:

[[OMEGA].sub.cr] = [[pi].sup.2]/4 x [square root of E/[rho]] x d/[l.sup.2] (7)

then the spindle vibrations remain stable.

3. CASE STUDY

Is considered a steel drill with d = 4 x [10.sup.-3] m, l = 0.1 m; [[OMEGA].sub.cr] = 5120 [s.sup.-1] corresponding to [n.sub.cr] = 48895 rpm.

The system is stable and initially the transversal vibrations on the two axes present maximum values:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

4. RESULTS AND CONCLUSIONS

In the example presented it was determined the critical angular speed for a d = 4 x [10.sup.-3] m drill. The next figure presents the variation of the critical angular speed as function of the diameter and length of the drill.

The dependency relation is:

[[OMEGA].sub.cr] = 12803 x d/[l.sup.2] (10)

where d [member of] (0.001/0.01)m; l [member of] (0.05/0.15)m

[FIGURE 1 OMITTED]

It can be observed that the critical speed increases proportionally with the drill diameter and decreases with its length square.

Using (8), (9) we can determine the amplitudes of transversal vibrations at the surface of the work material. For example, for a drilling process with l = 0.1 m, [l.sub.0] = 0.01 m, d = 0.01 m, F = 1550 N it is obtained:

[u.sub.max]([l.sub.0]) = [square root of [u.sup.2.sub.lmax] + [u.sup.2.sub.2max]] = 0.554 x [10.sup.-6]m (11)

Consequently, for this process the influence of transversal vibrations on the processing precision is insignificant.

5. REFERENCES

Huseyin, K.; Hagedorn, P. & Teschner, W. (1983). On the Stability of Linear conservative Gyroscopic Systems, In: Journal of Applied Mathematics and Physics, vol. 34, pp. 807-815.

Huseyin, K. (1991). On the Stability Criteria for Conservative Gyroscopic Systems, In: Journal of Vibration and Acoustic, vol 113, pp. 58-61.

Renshaw, A.A. & Mote, C.D. (1996). Local Stability of Gyroscopic Systems Near Vanishing Eigenvalues, In: ASME Journal of Applied Mechanics, vol. 63, pp. 116-120.

Shieh, R.C. (1971). Energy and Variational Principles for Generalized (Gyroscopic) Conservative problems, In: International Journal of Non-Linear Mechanics, vol. 5, pp. 495-509.

Wickert, J.A. & Mote, C.D. (1990). Classical Vibration Analysis of Axially Moving Continua, In: ASME Journal of Applied Mechanics, vol. 57, pp. 738-744.