Considerations about the different behaviors of the hydrostatic and hydrodynamic bearings using non-linear complex models.

Motomancea, Adrian

Abstract The field of bearings for machine-tools including here hydrostatic and hydrodynamic bearings is very important in connection with the importance of this stuffs in assuring the precision of the machine-tools. That is why we present here some researches in creating of non-linear complex models for both hydrostatic and hydrodynamic types of bearings, some methods of study of stability and some numerical results. We found zones of "order" where the bearing works in normal parameters but even the possibility of working in chaotic zones when some of the parameters are out of order.

Key words: bearing, stability, chaotic behavior, pressure champ.

1. INTRODUCTION

It is well-known that in the case of hydrostatic bearings the tangential stress of the fluid lubricates film is determinative for the forces and moment evaluation. In the case of hydrodynamic bearings the pressures-champ is necessary to be coupling over the tangential-stress. But really the both types of bearings work mixed. We present in this paper some aspects of this problem based on (Motomancea 1998), (Motomancea&Bugaru 2000), (Motomancea et. al., 2002) resuming two original non-linear complex models of both types of bearings, studying their dynamical behavior, finding some values of critical damping coefficient and some zones of chaotic behavior. The models are elaborated after very complex mathematical calculus and include a lot of parameters that influence the shaft behavior. Usually the existing models describe the shaft movement by two differential equations by 2nd order but in this paper it is used a new equation from kinetic moment theorem. This new equation improves the model, giving a more realistic one, the results of it being in high part in accordance with practical measurements.

It is very interesting that it was find in any circumstances of bearing work, some chaotic behavior. In order to validate these results, it was elaborating a method for the calculus of Lyapunov's maximum coefficient in a four-dimension space.

In the future the research is guiding on elaborated of an algorithm for the calculus of four Lyapunov's coefficients and to explain what means this, in a four-dimension space.

It is well-known that in the case of hydrostatic bearings the tangential stress of the fluid lubricates film is determinative for the forces and moment evaluation. In the case of hydrodynamic bearings the pressures-champ is necessary to be coupling over the tangential-stress. But really the both types of bearings work mixed.

In the case of hydrostatic bearings the tangential stress is given (in accordance with Newton's theory) by the next formula:

[tau] = [mu] v/h (1)

[tau] = [mu] v/h [+ or -] h/2 [absolute value of [partial derivative]p/[partial derivative]x] (2)

where: t-tangential stress [micro]--dynamic viscosity
 v-velocity h-film thickness
 p-pressure x-longitudinal coordinate

For the cylindrical radial hydrodynamic bearings it is possible to consider that the coupling (t-p) is given by:

(3)

In the case of hydrodynamic bearings the expression of tangential stress is correct by a term proceeding from the pressures champ:

The difference between (1) and (2) formulas conduce to different movement equations in these cases.

2. THE MOVEMENT EQUATIONS IN THE CASE OF HYDROSTATIC BEARINGS

In according with [1] we obtained the next system of non-linear differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

namely a four differential equations system, non-linear by first order, having: r, v, w, [OMEGA] variables and p, C, [alpha], [delta] and [[OMEGA].sub.0] parameters, where it's noted:

[??] = v and r[??] = r[omega] = w [??] [??] = r[??] + [??][??] = r[??] + [??][omega] (5)

In this system, R is the radius of the shaft, [delta] the film thickness, [alpha] includes damping, p includes stiffness, C includes dynamic viscosity coefficient, [OMEGA]0 includes the initial angular speed of the shaft, r is the radial displacement of the shaft center and [theta] the variable angle of r.

We are demonstrating that the system's equilibrium solution is unstable because the proper values determinative has a null root.

It was also demonstrate that the equation's system did not admit periodic solutions and a volume of a domain applies through phase flow, decrease.

For a hydrostatic bearing having the diameter D = 50 mm, the shaft's length L = 100 mm, the mass M = 3 kg and running between Nmin = 1 rpm and Nmax = 2500 rpm we tried a lot of numerical integration using Runge-Kutta integrator of IVth order from MATLAB utility, alternating all the parameters of the system in the normal running limits of the bearing but even in the outer of these.

3. MOVEMENT EQUATIONS IN THE CASE OF HYDRODYNAMIC BEARINGS

The pressure term appeared in (3) formula is:

dp/d[theta] = [mu][omega][R.sup.2]/[h.sup.2] (1 - [h.sub.0]/h) (6)

[theta]-angular coordinate R-shaft radius

[omega]-angular velocity of the shaft

[h.sub.0]-film thickness for [partial derivative]p/[partial derivative]x = 0.

In according with [2] we obtained the next system of nonlinear differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where f ([bar.r],[bar.[OMEGA]]) is a non-linear function having [bar.r] and [bar.[OMEGA]] variables and C and R parameters.

It is easy to observe that the single difference between (7) and (4) is in the fourth equations given by the f ([bar.r],[bar.[OMEGA]]) term. We used the same type of bearing and the same values for parameters, trying a lot of numerical integrations and we compared the results in these two cases.

4. RESULTS AND CONCLUSIONS

We obtained some graphic images of the dependence v = v (r) and [OMEGA] = [OMEGA](r), finding all the possible situations: limit point, limit cycle, limit torus and even strange attractors. It was surely determine two zones where the chaos has really chances to produce.

One, in the rotation zone of N = 30-40 rpm, other on the neighborhood of rotation N = 2500 rpm, for low damping but not missing (Motomancea et al., 1999).

Next we present the representative images obtained for certain values of the parameters.

We elaborate a method of calculus for maximum Lyapunov's exponent in order to certify the two found zones of chaotic movement.

The positive value of this coefficient in the case of chaotic movement is a validation for this type of movement.

We used the same values for the parameters and initial conditions in order to compare the graphic images between the situation without pressure champ and the situation with f ([bar.r],[bar.[OMEGA]]) providing from the pressure champ.

The results are comparable; namely the periodic movement and chaotic movement are present in both situations approximately in the same conditions. However we found a very interesting situation.

There are some zones of transition between regular behavior and chaotic behavior found in (Motomancea 1998) which in (Motomancea &Bugaru 2000) present a clear aspect of strange attractor. Fig. 5 presents the found situation without f ([bar.r],[bar.[OMEGA]]) term and fig. 6 presents the new situation with this term.

[FIGURES 1-6 OMITTED]

5. REFERENCES

Motomancea, A. (1998). Chaotic behaviors in hydrostatic bearings of the machine-tools, International Conference on Manufacturing Systems, ICMAS 98, Politehnica University of Bucharest, Machine and Production Systems Dept., TCMM 33, Editura Tehnica, ISBN 973-31-1236-4, pp.79-86.

Motomancea, A., Bugaru, M. (2000). The influence of the pressures champ on the chaotic behavior of the hydrostatic bearings, T.C.M.M., Bucharest, October 2000, ISBN 973-31-1492-8(6), Vol. 40, pp.89-94.

Motomancea, A., Bugaru, M., Dragomirescu, C., Predoi, M., (1999). A non linear-model for a hydrostatic bearing. Chaotic behaviors found in two different regions of rotation, Sixth International Congress on Sound and Vibration, Denmark, Copenhagen, 5-8 July, pg.3245-3252.

Motomancea, A., Dragomirescu, C., Predoi, M., Bugaru, M., (1999). New-non-linear model on hydrostatic bearings. The study of the instability zone, European Conference on Computational Mechanics ECCM '99, Germany, Munchen, Aug. 31-Sept. 3. published on CD-ROM, with the resume in ABSTRACTS, pp. 578.

Motomancea, A., Cotet, C., Bugaru, M., Enescu, N.,(2002) Order and Chaos in Machine-Tools Bearings, GRACM 2002 4-th GRACM Congress on Computational Mechanics, Patras, Greece, 27-29 June 2002.