Mathematical model for dynamic analysis of the contact between tool and workpiece at high-speed grinding.

Popescu, Daniel ; Bolcu, Dumitru

1. INTRODUCTION

The vibrations that accompany the grinding process are complex, being caused by a large number of factors that participate in the manufacturing process, making it difficult to obtain finished surfaces with high precision and quality in cost-effective conditions (Ispas & Popescu, 1998).

The vibrations that occur in normal direction on the processed surface are considered to be reduced due to the rigidity between tool and workpiece.

In this case it is possible the forming of waves on the tool edge, due to vibrations caused by the unbalance of the tool fixing system, but also of waves that occur during straightening (Toushoff et. Al., 1992).

The results obtained from the designed model provide an fairly accurate image of the process. The proposed model is intended to determine the influence of the cutting speed on the magnitude of vibrations, and thus the influence upon the dimensional precision ard surface finish quality (Schultz & Toshimichi, 1992), (Chen & Wang, 1994).

2. ESTABLISHMENT OF THE ANALYSIS MODEL

It is considered the schematic presented in fig. 1:

[FIGURE 1 OMITTED]

In order to determine the dependence between the input measurements, i.e. cutting force components, and the output measures, namely the relative displacements between tool and workpiece, the general form Lagrange relations are employed.

d/dt([delta]T/[delta][[??].sub.j] - [delta]T/[delta][q.sub.j] = [Q.sub.j],; j = [bar.1,10]) (1)

Where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Relation (2) gives the total kinetic energy of the system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Relation (3) represents the generalized forces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In which:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--indicates the influence of forces upon movement

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--indicates the influence of torques upon movement

Considering the definition of the elastic potential and dissipation potential:

V = 1/2k x [DELTA][l.sup.2] = 1/2k x [chi square]

V = 1/2k x [DELTA][l.sup.2] = 1/2k x [([x.sub.1] - [x.sub.2]).sup.2] (4)

We have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

By computing the terms of(1) corresponding to each coordinate axis and replacing into the energy equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Remark: The initial reference system is positioned with the origin on the tool axis, with x-axis along the tool bearing spindle; the y-axis crosses the point of contact between tool and piece while the z-axis is perpendicular on the plane defined by other two axes, forming a tri orthogonal normal system.

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The differential equation that describes the vibration of the system is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The meaning of [M], [C], [K] and [F] is given in (Popescu, 1999).

Remarks: The cutting force components on the three coordinate axes can be determined using:

[F.sub.x] = [F.sub.0][lambda] cos [omega]t cos [alpha]

[F.sub.y] = [F.sub.0][lambda] cos [omega]t sin [alpha]

[F.sub.z] = sin [omega]t (10)

In which:

[F.sub.0]--nominal cutting force

[lambda]--overlapping factor between actual and previous tool pass

[omega]--disturbance force throb

[alpha]a--angular position of [F.sub.0] with respect to the system [F.sub.x], [F.sub.y], [F.sub.z].

The forced solution of equation (4) has the form:

{q} = {A}sin [omega]t + {B}cos [omega]t (11)

In which:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

It results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

3. CONCLUSION

--Vibration amplitudes increase with the value of the product between [F.sub.0] and [lambda].

--[q.sub.x1], [q.sub.x2], [q.sub.[theta]1], [q.sub.[theta]2] increase with cos([alpha]) (when [alpha] decreases).

--[q.sub.y1], [q.sub.y2] increase with sin([alpha]) (when [alpha] increases).

--[q.sub.z1], [q.sub.z2] are independent of [alpha].

--The vibration amplitudes fade over time.

--It is preferred a high cutting speed since it results in low vibration amplitudes, which leads to increased processing accuracy and finished surface quality.

4. REFERENCES

Chen, C. H. & Wang, K. W. (1994). An Integrated Approach Toward The Dynamic Analysis Of High-Speed Spindles System Model, Journal Of Vibration And Acoustics, pg. 514-522, ISSN 1048-9002

Ispas, C. & Popescu, D. (1998). Mathematical Model For Analysis Of Dependencies Between Main Components Of Dynamic System At Internal Grinding Machines, 8th International Conference On Managerial And Technological Engineering "TEHNO 98", Timisoara, Romania, ISBN 973-0-00596-6

Popescu, D. (1999). Theoretical And Experimental Contributions Regarding Improvement Of Processing Precision At Internal Grinding Machines, PhD Thesis, Bucharest

Toushoff H.K., Peters J., Inasaki I. & Paul T. (1992). Modeling And Simulation Of Grinding Process, Anals of CIRP, vol.41/2/1992, ISSN 0007-8506

Schultz, H. & Toshimichi M. (1992). High Speed Machining, Annals of CIRP, 41/2/1992, ISSN 0007-8506